J.evol.equ. 4 (2004) 75 – 971424–3199/04/010075 – 23DOI 10.1007/s00028-003-0079-x© Birkhauser Verlag, Basel, 2004

Asymptotic properties of solutions of the viscousHamilton-Jacobi equation

Piotr Biler, Mohammed Guedda and Grzegorz Karch

Abstract. The purpose of the paper is to study properties of solutions of the Cauchy problem for the equationut −�u+|∇u|q = 0 under the assumption (n+2)/(n+1) < q < 2. General selfsimilar solutions are constructed.Moreover, for initial data with some decay at infinity, we determine the leading term of the asymptotics of solutionsin Lp(Rn) which is described by either solutions of the linear heat equation or by particular selfsimilar solutionsof the original equation.

1. Introduction

In this paper, we study the existence and qualitative properties of nonnegative solutionsof the Cauchy problem for the viscous Hamilton-Jacobi equation in (x, t) ∈ R

n × (0, ∞)

ut − �u + |∇u|q = 0, (1.1)

u(x, 0) = u0(x) ≥ 0, (1.2)

under the crucial assumption n+2n+1 < q < 2.

Equations of that kind appear as the viscosity approximation to the first order partialdifferential equations of Hamilton-Jacobi type, in the stochastic control theory, as well asin a number of interesting and different physical considerations. The first example of anequation of the type (1.1) in one space dimension with 1 ≤ q ≤ 2 is the well knownKardar-Parisi-Zhang equation from the theory of growth and roughening of surfaces, cf.e.g., [22, 15, 17] and the references therein. These equations appear also in a probabilisticcontext, see e.g. [7]. The above Cauchy problem (1.1)–(1.2) for different q’s has beeninvestigated recently by many people. We recall (the list of references is by no meansexhaustive) the papers of Ben-Artzi and his collaborators [1, 8, 9, 10], Weissler and hiscoauthors [9, 13, 26, 27] (the analysis focused on the invariance properties of equations),and a novel approach (based on Bernstein type estimates) in recent works of Benachour andLaurencot [3, 4, 5, 23], cf. also [15].

Received 23 January 20022000 Mathematics Subject Classification: 35B40, 35K55, 35Q99.Key words and phrases: Viscous Hamilton-Jacobi equation, selfsimilar solutions, asymptotic behavior.

76 piotr biler, mohammed guedda and grzegorz karch J.evol.equ.

The threshold value of the exponent q = n+2n+1 is not only important in the existence of

solutions questions (under quite weak regularity assumptions on the initial data which canbe bounded measures, cf. [3]) but mainly for the analysis of large time behavior of solutions.Namely, if 1 < q ≤ n+2

n+1 the L1-norm of a solution u(t) (mass) decays to 0 as t → ∞,

and if q > n+2n+1 , mass does not vanish and reaches a nonzero limit: limt→∞ ‖u(t)‖1 > 0

([10, 3]).Some other results concerning the decay estimates of solution, their extinction (the

vanishing identically after a finite time), behavior for large q, etc., can be found in recentpapers [6, 23]. Preliminary results on the large time behavior of solutions of systems ofequations of the type (1.1) are proved in [2]. The large time behavior of the L1-norm ofsolutions of equations more general than (1.1) can be found in [25].

Note that for q ≥ n+2n+1 there is no source solution of (1.1), i.e. a solution with u0 = Mδ0,

M > 0, see [3, Th. 4]. Such a (fundamental) solution is usually of great importance ina study of nonlinear equations. Thus, for q > n+2

n+1 one may expect quite a complexasymptotics of solutions of (1.1), cf. a similar situation for convection-diffusion equationsin, e.g., [20, 11].

One of the goals of this paper is the construction of selfsimilar solutions. It is straight-forward to check that if u(x, t) is a solution of Equation (1.1) then so is uλ(x, t) =λau(λx, λ2t) for a = (2−q)/(q −1) and every λ > 0. A solution satisfying the invarianceproperty

λau(λx, λ2t) ≡ u(x, t) (1.3)

for every x ∈ Rn, t > 0 and λ > 0, is called the (forward) selfsimilar solution.

The selfsimilar very singular solutions of (1.1) for 1 < q < (n + 2)/(n + 1) wereconstructed by Benachour and Laurencot [4, 5] as suitable limits of source solutions of(1.1). On the other hand, if (n + 2)/(n + 1) < q < 2 and p = q/(2 − q) the problem ofexistence of forward selfsimilar solutions of a more general equation

ut = �u + a|∇u|q + b|u|p−1u (1.4)

with a, b ∈ R was studied by Snoussi et al. [26]. Under the assumption that the quantity

supt>0

{tβ‖et�u0‖r , t1/2+β‖∇et�u0‖r} with β = 2 − q

2(q − 1)− n

2r,

is finite and sufficiently small, the solution corresponding to u0 is global in time and uniquein a suitable function space. For u0 homogeneous of degree −a one obtains a family ofselfsimilar solutions. Several other results on selfsimilar solutions of (1.4) can be foundin [28].

We summarize our results in Section 2. In Section 3, we prove the existence of generalselfsimilar solutions of the Equation (1.1). Section 4 deals with the asymptotic properties of

Vol. 4, 2004 Viscous Hamilton-Jacobi equation 77

(radially symmetric) selfsimilar solutions of (1.1). In Section 5, a result on weakly nonlinear(i.e. the same as for the heat equation) behavior of solutions with nonintegrable initial datasatisfying, roughly speaking, u0(x) ∼ |x|−b as |x| → ∞ with a = 2−q

q−1 < b < n, is

proved. Section 6 deals with integrable nonnegative initial data 0 ≤ u0 ∈ L1(Rn) whichlead to a simple large time asymptotics of solutions determined by the heat kernel. Namelyu(t) ∼ M∞G(., t), where

G(x, t) = (4πt)−n/2 exp

(−|x|2

4t

), (1.5)

and M∞ = limt→∞∫

Rn u(x, t) dx is finite and strictly positive provided q > n+2

n+1 .For 1 ≤ p ≤ ∞ the Lp(Rn)-norm of a Lebesgue measurable real-valued function

defined on Rn is denoted by ‖f ‖p. The letter C stands for generic positive constants which

may vary from line to line during considerations.

2. Results and comments

The theory on the well-posedness of solutions of the Cauchy problem (1.1)–(1.2) underthe only assumption u0 ∈ L∞(Rn) ∩ C(Rn) (no nonnegativity assumption) is developedby Gilding et al. in [15]. They prove that each solution corresponding to such an initialdatum is unique, global and classical. That paper extends previous results by Amour andBen-Artzi [1] where nonnegative initial conditions belonging to L1(Rn)∩C2(Rn)∩W 2,∞(Rn) have been considered. We also refer the reader to [3, Prop. 5] for similar results withu0 ∈ L1(Rn)∩Lq(Rn), and to [25, Th. 1] for a theory concerning more general equations.

In our first theorem, we present a general method of construction of selfsimilar solutionsof (1.1) of the form (1.3). Our reasoning requires the following assumption

n + 2

n + 1< q < 2 (2.1)

which is equivalent to

a = 2 − q

q − 1∈ (0, n). (2.2)

THEOREM 2.1. Assume (2.1). Let u0(x) be a nonnegative homogeneous functionof degree −a, continuous on R

n \{0}. Denote by uK(x, t) the unique classical solutionof (1.1)–(1.2) with the initial datum uK

0 (x) = min{u0(x), K}. The family of functionsuK(x, t) converges as K ↗ ∞ for every (x, t) ∈ R

n × (0, ∞) toward a selfsimilar solu-tion U(x, t) of equation (1.1). This selfsimilar function satisfies: U(·, t) ∈ Lp(Rn) forevery t > 0 and p ∈ (n/a, ∞].

Applying the Lebesgue Dominated Convergence Theorem we improve the pointwiseconvergence of uK to U in the following corollary.

78 piotr biler, mohammed guedda and grzegorz karch J.evol.equ.

COROLLARY 2.2. Under the assumptions of Theorem 2.1,

‖uK(·, t) − U(·, t)‖p → 0 as K ↗ ∞ (2.3)

for every p ∈ (n/a, ∞] and every t > 0. Moreover, for each fixed K > 0,

t (a−n/p)/2‖uK(·, t) − U(·, t)‖p → 0 as t → ∞ (2.4)

for every p ∈ (n/a, ∞].

Note that using the selfsimilar form of U(x, t) = t−a/2U(x/√

t, 1) we have

t (a−n/p)/2‖U(·, t)‖p = ‖U(·, 1)‖p.

Thus, the relation (2.4) says that the large time behavior in Lp(Rn) of the solution uK(x, t)

is described by the selfsimilar solution U(x, t). The next corollary shows that this propertyis enjoyed by a large class of solutions of the problem (1.1)–(1.2).

COROLLARY 2.3. Assume that u0, uK0 , and U satisfy the conditions in Theorem 2.1.

Let v0 ∈ L∞(Rn) ∩ C(Rn) satisfy uK10 (x) ≤ v0(x) ≤ u

K20 (x) for all x ∈ R

n and someconstants 0 < K1 < K2. Denote by v(x, t) the solution of (1.1) with v0 as the initialdatum. For every p ∈ (n/a, +∞],

t (a−n/p)/2‖v(·, t) − U(·, t)‖p → 0 as t → ∞. (2.5)

The proofs of Theorem 2.1 and Corollaries 2.2, 2.3 are given in Section 3. Here, weonly mention that it is possible to obtain selfsimilar solutions of the Equation (1.1) usingapproximations different from those in Theorem 2.1. For example, one can consider thefamily of solutions uε(x, t) of (1.1) with the initial data uε

0(x) ≡ A(ε+|x|)−a . An argumentcompletely analogous to that in the proof of Theorem 2.1 gives uε(x, t) ↗ U(x, t) as ε ↘ 0,where U is a selfsimilar solution of (1.1). Moreover, choosing a continuous initial datumv0(x) such that A(ε1 + |x|)−a ≤ v0(x) ≤ A(ε2 + |x|)−a for some ε1 > ε2 > 0, we seethat the solution v(x, t) corresponding to v0 satisfies the asymptotic relation (2.5).

For radially symmetric u0, the selfsimilar function U(x, t) from Theorem 2.1 is radiallysymmetric with respect to x as the pointwise limit of radially symmetric functions uK .Hence, it is easy to deduce from (1.3) that

U(x, t) = t−a/2f

( |x|√t

)≡ t−a/2f (ξ), (2.6)

where f (ξ) = U((ξ, 0, . . . , 0), 1), ξ ∈ (0, ∞). Our next goal is to study properties of theprofile f in (2.6) using a classical approach via nonlinear ordinary differential equations.If we substitute (2.6) in (1.1) we obtain the second order ordinary differential equation

f ′′ + n − 1

ξf ′ − |f ′|q + a

2f + 1

2ξf ′ = 0, (2.7)

Vol. 4, 2004 Viscous Hamilton-Jacobi equation 79

where ′ =d/dξ . Sincef (−ξ) solves (2.7) as well, we may impose, without loss of generality,the boundary conditions

f (0) = β, f ′(0) = 0, (2.8)

for some β > 0. The problem (2.7)–(2.8) has a unique maximal solution f = f (., β) ∈C2[0, ξmax), and this fact can be proved following the reasoning in [13, Prop. 4.4]. Theaim of the next theorem is to study a possible extension of f and its asymptotic behavioras ξ → ∞.

THEOREM 2.4. Assume that n+2n+1 < q < 2, a = (2 − q)/(q − 1), and β > 0. The

unique local solution f of (2.7)–(2.8) is global, nonnegative, decreases to 0 as ξ tends toinfinity, and satisfies

f (ξ) = Lξ−a

{1 + A

ξ2+ o

(1

ξ2

)}, as ξ → ∞, (2.9)

where L = L(β) > 0 and A = a2 − (n − 2)a − aqLq−1.

REMARK 2.5. Phenomena analogous to those described in Theorem 2.4 have beenobserved for other equations in several recent papers. Here, let us only recall the work byBrezis et al. [12] on very singular selfsimilar solutions of the equation ut = �u − up.Similar results concerning other problems can be found, e.g., in [24, 11, 15, 16].

REMARK 2.6. Here, we would like to emphasize that the asymptotic behavior for largeξ (in the first approximation) of the solution f from Theorem 2.4 is that for the linearizedequation f ′′ + (n − 1)ξ−1f ′ + a

2 f + 12ξf ′ = 0, and this property is independent of the

range of β. There are several examples where the existence of nonnegative solutions andtheir asymptotic behavior depend in a very sensitive way on the form of the nonlinear termin the equation under consideration, and on the initial condition f (0). Such a situation forvarious convection-diffusion problems is described in, e.g., [24, 11, 16, 17], and for otherexamples of quasilinear equations-in [27, 28].

REMARK 2.7. For 1 < q < (n + 2)/(n + 1), Benachour and Laurencot proved in[4, 5] the existence of a solution f of (2.7) decaying faster than in (2.9). In that case,f ∈ L1(R+; rn−1dr) ∩ C∞(0, ∞), f ′(0) = 0 and ξaf (ξ) → 0 as ξ → ∞.

REMARK 2.8. Equation (2.7) has particular solutions of the form f (ξ) = Lξγ forsome L > 0 and γ ∈ R. Indeed, after substituting this function in (2.7) we obtain γ = −a

and aq−1Lq−1 = (n − q(n − 1))/(q − 1). Hence, for 1 < q < n/(n − 1) such a solutionexists and has a singularity at the origin. On the other hand, for q > 2 and n �= 1, thesolution f (ξ) = Lξ−a exists for all ξ ≥ 0, and is continuous at the origin, because−a = −(2 − q)/(q − 1) > 0.

80 piotr biler, mohammed guedda and grzegorz karch J.evol.equ.

REMARK 2.9. Let us compare Theorem 2.4 with an analogous result for the Equation(2.7) with q = 2 (hence, a = 0). Using standard arguments similar to those in [13], one canshow that the function f (ξ) ≡ β is the only solution of the equation

f ′′ + n − 1

ξf ′ − (f ′)2 + 1

2ξf ′ = 0 (2.10)

satisfying the boundary conditions (2.8). Moreover, the Equation (2.10) is integrable aslong as we assume that f ′(ξ) �= 0 for every ξ ≥ 0. Indeed, a direct calculation showsthat the function v(ξ) = f ′(ξ)eξ2/4ξn−1 satisfies the equation v′ − ξ1−ne−ξ2/4v2 = 0.

A simple integration of this equation leads to v(ξ) = (c1 − ∫ ξ

1 τ 1−ne−τ 2/4 dτ)−1. Finally,

we integrate the equation f ′(ξ) = ξ1−ne−ξ2/4v(ξ) to obtain the two-parameter family ofnonconstant solutions of (2.10) given by

f (ξ) = c2 − log

(c1 −

∫ ξ

1τ 1−ne−τ 2/4 dτ

). (2.11)

Now, we can derive asymptotic properties of the selfsimilar profile f defined in (2.11).For example, for c1 >

∫ ∞1 τ 1−ne−τ 2/4 dτ the solution of (2.10) is defined on (0, ∞),

and satisfies limξ→∞ f (ξ) = c2 − log(c1 − ∫ ∞1 τ 1−ne−τ 2/4 dτ). On the other hand, for

c1 <∫ ∞

1 τ 1−ne−τ 2/4 dτ there exists ξ0 > 0 such that f tends to ∞ as ξ ↗ ξ0.

It follows from Theorem 2.4 that for each β > 0, the corresponding solution f = fβ of(2.7)–(2.8) satisfies

limξ→∞ ξafβ(ξ) = L(β) and lim

ξ→∞ f ′β(ξ) = 0, (2.12)

where L(β) > 0. Our next result assures the uniqueness of solutions satisfying (2.12).

PROPOSITION 2.10. The constant L = L(β) in (2.12) is strictly increasing as afunction of β > 0.

The proofs of Theorem 2.4 and Proposition 2.10 are contained in Section 4.In the remainder of this section, we present our results on the large time behavior of

solutions of the initial value problem (1.1)–(1.2). Preliminary results in this direction arealready included in Corollaries 2.2 and 2.3. They say, roughly speaking, that if u0 behavesfor large |x| like a homogeneous function of degree −a, then the large time behavior isdescribed by a selfsimilar solution. Now, we are going to consider initial conditions whichtend to 0 as |x| → ∞ faster, namely, our standing assumption is

0 ≤ u0(x) ≤ B(1 + |x|)−b (2.13)

for all x ∈ Rn, a constant B > 0, and

b > a = 2 − q

q − 1for

n + 2

n + 1< q < 2. (2.14)

Vol. 4, 2004 Viscous Hamilton-Jacobi equation 81

We show that under these conditions the large time behavior of solutions of (1.1)–(1.2) isweakly nonlinear, i.e., solutions behave for large t like solutions of the heat equation.

THEOREM 2.11. Assume (2.14) and a < b < n. Let u(x, t) be the solution of(1.1)–(1.2) with the initial datum u0 satisfying (2.13). Then, for each p ∈ (n/b, ∞] thefollowing relation

‖u(·, t) − et�u0‖p = o(t−b/2+n/(2p)) as t → ∞ (2.15)

holds true.

Here, we stress on the fact that the proof of Theorem 2.11 requires optimal estimates of‖∇u(·, t)‖p for every p ∈ (n/b, ∞] which we obtain in Proposition 5.2 below. Those esti-mates applied to the following integro-differential equation (satisfied by every sufficientlyregular solution of (1.1)–(1.2))

u(t) = et�u0 −∫ t

0e(t−τ)�|∇u(τ)|q dτ (2.16)

lead, almost immediately, to the relation (2.15). Recall that et�u0 = G(t) ∗ u0, where theheat kernel G(x, t) is defined in (1.5).

REMARK 2.12. Theorem 2.11 holds true under weaker assumptions on u0 than thatformulated in (2.13). In fact, we only need to assume that ‖et�u0‖p ≤ C(1+ t)−b/2+n/(2p)

for every p ∈ (n/b, ∞], all t > 0, and C independent of t .

REMARK 2.13. Assume that u0 satisfies (2.13) and, moreover, lim|x|→∞ |x|bu0(x) =K for some K > 0. It is well known (cf. e.g. [18, Lemma 3.3]) that, in this case, we have

‖et�u0 − Kgb(·, t)‖p = o(t−b/2+n/(2p)) as t → ∞

for every p ∈ (n/b, ∞], where the function gb(x, t) = t−b/2gb(|x|/√t, 1) is the uniqueselfsimilar solution of the heat equation ut = �u with the initial datum u(x, 0) = |x|−b.Combining this fact with (2.15) we obtain that the large time behavior of the solution u(x, t)

of the Cauchy problem (1.1)–(1.2) is described by Kgb(x, t), too.

Finally, we study the asymptotic behavior in Lp(Rn) for every p ∈ [1, ∞] of solutionscorresponding to integrable initial conditions. Here, our results are valid for every q > 0and sufficiently regular solutions of (1.1)–(1.2), however, they are optimal as long as q >

(n+2)/(n+1). Indeed, it is well known (cf. [1, 10]) that the constant M∞ in Theorem 2.14below is equal to 0 for q ≤ (n + 2)/(n + 1).

82 piotr biler, mohammed guedda and grzegorz karch J.evol.equ.

THEOREM 2.14. Assume that u(x, t) is the unique nonnegative solution of the Cauchyproblem (1.1)–(1.2) corresponding to the nonnegative initial datum u0 ∈ L∞(Rn) ∩C(Rn) ∩ L1(Rn). If q > (n + 2)/(n + 1), the large time behavior of u in L1(Rn) isdescribed by a multiple of the heat kernel, namely,

‖u(·, t) − M∞G(·, t)‖1 → 0 as t → ∞, (2.17)

where the constant

M∞ = limt→∞

∫R

nu(x, t) dx =

∫R

nu0(x) dx −

∫ ∞

0

∫R

n|∇u(x, t)|q dx dt

is finite and positive.

Theorem 2.14 is a direct consequence of Theorem 6.1 from Section 6 where we studythe Cauchy problem for the linear nonhomogeneous heat equation ut = �u + f with anintegrable initial condition u0 and f = f (x, t). In that case, the L1(Rn)-asymptotics isgiven by the function M∞G with M∞ = limt→∞

∫R

n u(x, t) dx and the heat kernel G

in (1.5).The asymptotic behavior of solutions in Lp(Rn) with p �= 1 is now deduced directly

from Theorem 2.14.

COROLLARY 2.15. Under the assumptions of Theorem 2.14, for every p ∈ [1, ∞]

tn(1−1/p)/2‖u(·, t) − M∞G(·, t)‖p → 0 as t → ∞. (2.18)

For the proofs of Theorem 2.14 and Corollary 2.15, we refer the reader to Section 6.

3. Selfsimilar solutions

In this section, we prove Theorem 2.1 and Corollaries 2.2, 2.3.

Proof of Theorem 2.1. For every K > 0, the function uK0 (x) = min{u0(x), K} is continuous

and bounded. Hence, the solution uK exists and is unique by the construction in [15].Moreover, by the comparison principle, we immediately obtain that if 0 < K1 < K2, thenuK1(x, t) ≤ uK2(x, t) for every (x, t) ∈ R

n × [0, ∞). Thus,

uK(x, t) is increasing as K ↗ ∞ (3.1)

for all (x, t) ∈ Rn×[0, ∞). This family is also uniformly bounded from above by a function

independent of K , what results from the following inequalities

0 ≤ uK(x, t) ≤ (G(·, t) ∗ uK0 )(x) ≤ B(G(·, t) ∗ | · |−a)(x), (3.2)

Vol. 4, 2004 Viscous Hamilton-Jacobi equation 83

where G is the heat kernel (1.5). The first two inequalities in (3.2) are a simple consequenceof the comparison principle. The third inequality in (3.2) is obtained using the elementaryestimates

0 ≤ uK0 (x) ≤ u0(x) ≤ B|x|−a with B = sup

x∈Rn\{0}

u0(x/|x|).

Here, it is worth emphasizing that |x|−a /∈ Lp(Rn) for any p ∈ [1, ∞], but

G(·, t) ∗ | · |−a ∈ Lp(R) for each p ∈ (n/a, ∞] (3.3)

and all t > 0. Indeed, by (2.2), the fact formulated in (3.3) is a direct consequence of theHardy-Littlewood-Sobolev inequality [19, Th. 4.5.3 and Lemma 4.5.4].

Combining (3.1), (3.2), and (3.3) we establish the existence of a function U(x, t) suchthat U(·, t) ∈ Lp(Rn) for each p ∈ (n/a, ∞] and all t > 0; moreover,

uK(x, t) ↗ U(x, t) as K ↗ ∞ for all (x, t) ∈ Rn × [0, ∞). (3.4)

In the next step of our proof, we show that U(x, t) satisfies the Equation (1.1) in thesense of distributions. Here, the reasoning is similar to that in [3], hence we shall be briefin detail. First, let us recall the inequality from [3, p. 2003]

‖∇u(q−1)/q(·, t)‖∞ ≤ (q − 1)1/2‖u(·, t/2)‖(q−1)/q∞ t−1/2 (3.5)

which is valid for every sufficiently regular solution of (1.1)–(1.2) and all t > 0. Hence,applying (3.5) combined with (3.2) to uK, we get

‖∇(uK(·, t))(q−1)/q‖∞ ≤ Ct−a(q−1)/(2q)−1/2

for all t > 0 and a constant C independent of K and t .Now, using the identity ∇uK = q

q−1 (uK)1/q∇(uK)(q−1)/q , we obtain

‖∇uK(·, t)‖∞ ≤ q

q − 1‖uK(·, t)‖1/q∞ ‖∇(uK)(q−1)/q‖∞ (3.6)

≤ Ct−a/2−1/2

for all t > 0 and another constant C independent of t and K . In particular, this implies thatthere exists a sequence Km ↗ ∞ such that

{|∇uKm |q} converges weakly to g ∈ L2loc(R

n × (0, ∞)).

Thus, recalling (3.4), we deduce that U(x, t) is a solution of

Ut − �U + g = 0 in D′(Rn × (0, ∞)).

84 piotr biler, mohammed guedda and grzegorz karch J.evol.equ.

Finally, repeating the reasoning from [3, pp. 2008–2009], we conclude that

g = |∇U |q a.e. in Rn × (0, ∞).

Now let us check that U(x, t) is a selfsimilar function. By a direct calculation involvingthe homogeneity of u0, we have

λauK0 (λx) = min{λaK, u0(x)} = uλaK

0 (x).

Hence, recalling that the Equation (1.1) is invariant under the rescaling λau(λx, λ2t), by theuniqueness of solutions of (1.1)–(1.2) with bounded and continuous initial data, we obtain

λauK(λx, λ2t) = uλaK(x, t)

for every x ∈ Rn, t ≥ 0 and λ > 0. Now, taking the limit as K ↗ ∞ in the above equality

we arrive at

λaU(λx, λ2t) = U(x, t)

for all (x, t) ∈ Rn × [0, ∞) and λ > 0, so U(x, t) is selfsimilar.

Proof of Corollary 2.2. If p ∈ (n/a, ∞), the limit in (2.3) is a direct consequence of theLebesgue Dominated Convergence Theorem after applying (3.2)–(3.4). The case p = ∞follows from the Gagliardo-Nirenberg inequality

‖uK(·, t) − U(·, t)‖∞ ≤ C‖uK(·, t) − U(·, t)‖1−n/rr ‖∇(uK(·, t) − U(·, t))‖n/r

r , (3.7)

valid for every r > n, and from the estimates for the gradient ∇uK analogous to thosein (3.6) with the L∞(Rn)-norm replaced by the Lr(Rn)-norm with r sufficiently large(cf. (5.4), below).

Now, let us prove (2.4). We already know from the proof of Theorem 2.1 and from (2.3)that for every K > 0 and every fixed t0 > 0 the family λauK(λ ·, λ2t0) = uλaK(·, t0) tendsto U(·, t0) in Lp(Rn) as λ → ∞. Moreover, changing the variables we have

‖λauK(λ ·, λ2t0) − U(·, t0)‖p = λa−n/p‖uK(·, λ2t0) − U(·, t0)‖p.

Hence, for λ = √t/t0 we obtain (2.4).

Proof of Corollary 2.3. By the comparison principle, we have uK1(x, t) ≤ v(x, t) ≤ uK2

(x, t) for all x ∈ Rn and t > 0. Hence, repeating the argument from Corollary 2.2, we get

‖λav(λ ·, λ2t0) − U(·, t0)‖p → 0 as λ → ∞,

which is equivalent to (2.5).

Vol. 4, 2004 Viscous Hamilton-Jacobi equation 85

4. Properties of selfsimilar profiles

In this section, we study the structure of radial selfsimilar solutions of Equation (1.1),and our goal is to prove Theorem 2.4 and Proposition 2.10. For computational reasons, it ismore convenient for us to study, instead of (2.7)–(2.8), the following second order nonlinearordinary differential equation

f ′′ + n − 1

ξf ′ − λ|f ′|q + a

2f + 1

2ξf ′ = 0, (4.1)

with the normalized boundary conditions

f (0) = 1, f ′(0) = 0, (4.2)

under the crucial assumptions

λ > 0 andn + 2

n + 1< q < 2.

REMARK 4.1. Let f be a solution of (4.1)–(4.2) with λ = βq−1. A trivial calculationshows that βf (ξ) is the solution of the original problem (2.7)–(2.8).

The proof of Theorem 2.4 is based on two lemmata. The first of them says that solutionsof (4.1) are global and tend to 0 at infinity.

LEMMA 4.2. Let n+2n+1 < q < 2 and a = (2−q)/(q −1). Assume that f is the solution

of (4.1)–(4.2) defined on the maximal interval of existence [0, ξmax) for some ξmax ≤ ∞.

Then ξmax = ∞, f is nonnegative, f ′ < 0, and both f , f ′ tend to 0 as ξ approaches ∞.

Proof. Since f ′(0) = 0 and f ′′(0) = − a2n

, it follows that f ′ < 0 and f > 0 on somesmall interval (0, ε), ε > 0. Assume that f has a positive local minimum at ξ0 > 0.Using the Equation (4.1) we deduce that f ′′(ξ0) = − a

2 f (ξ0) < 0, which is a contradiction.Consequently, we have f ′ < 0 as long as f > 0.

Now assume for contradiction that f changes the sign, and let ξ0 be the least zero of f.

Obviously, f ′(ξ0) ≤ 0. In fact, f ′(ξ0) < 0, because in the case f (ξ0) = f ′(ξ0) = 0, wewould obtain f ≡ 0. We rewrite Equation (4.1) as

f ′′ + 1

2(ξf )′ = λ|f ′|q +

(1

2− a

2

)f − n − 1

ξf ′ (4.3)

which implies

f ′(ξ0) = λ

∫ ξ0

0|f ′|q(ξ) dξ +

∫ ξ0

0

((1

2− a

2

)f (ξ) − (n − 1)

f ′(ξ)

ξ

)dξ.

86 piotr biler, mohammed guedda and grzegorz karch J.evol.equ.

Let us prove that the function

G(ξ) =(

1

2− a

2

)ξf (ξ) − (n − 1)f ′(ξ) (4.4)

is nonnegative for all ξ ∈ (0, ξ0] (evidently, this will show that f cannot vanish). This factis obvious if n = 1 (cf. (2.2)). For n > 1 we have G(ξ0) = −(n − 1)f ′(ξ0) > 0. Supposethat there exists ξ1 ∈ (0, ξ0) such that G(ξ1) = 0, i.e.(

1

2− a

2

)ξ1f (ξ1) = (n − 1)f ′(ξ1), (4.5)

and G(ξ) > 0 in (ξ1, ξ0]. We integrate (4.3) over [ξ1, ξ0] to obtain

f ′(ξ0) − f ′(ξ1) − 1

2ξ1f (ξ1) = λ

∫ ξ0

ξ1

|f ′|q(ξ) dξ +∫ ξ0

ξ1

G(ξ)

ξdξ.

Thus, by the assumptions on G and ξ0, ξ1, we get f ′(ξ0) > f ′(ξ1) + 12ξ1f (ξ1). Finally,

using (4.5), we have the equivalent form of this inequality

f ′(ξ0) >

(1

n − 1

(1

2− a

2

)+ 1

2

)ξ1f (ξ1) = q(n − 1) − (n + 2)

2(q − 1)(n − 1)ξ1f (ξ1).

The right hand side is strictly positive for q > n+2n+1 , so this is a contradiction. Hence

G(ξ) > 0 for all ξ ∈ (0, ξ0] which finishes the proof that f remains positive on [0, ∞).Thus f is a nonnegative and strictly decreasing function. Assume that limξ→∞ f (ξ)> 0.

Setting H = (f ′)2 + af 2, and using (4.1) we immediately deduce that H is a monotonedecreasing function with a finite limit at infinity. Therefore f ′(ξ) → 0 as ξ → ∞. Thisfact combined with (4.1) leads to the inequality

f ′′ + 1

2ξf ′ ≤ − < 0,

valid for all large ξ and some > 0. Multiplying this inequality by eξ2/4 and integratingwe obtain

f ′(ξ) ≤ e−ξ2/4(

C −

∫ ξ

0eτ 2/4 dτ

)(4.6)

for some C > 0. It is easy to see that

limξ→∞

∫ ξ

0 eτ 2/4 dτ

1ξeξ2/4

= 2

(e.g. using either the l’Hopital rule or the inequality (ξ/2)∫ ξ

0 eη2/4 dη ≥ eξ2/4 − 1 validfor all ξ ≥ 0), which implies that f ′(ξ) ≤ −/ξ for large ξ . The last inequality impliesthat f tends to −∞ as ξ → ∞, which is a contradiction. �

Vol. 4, 2004 Viscous Hamilton-Jacobi equation 87

Now, by a reasoning similar to that in [12, 11, 16] we show that 〈f, f ′〉 approaches 〈0, 0〉along exactly one direction in the phase plane.

LEMMA 4.3. Assume that n+2n+1 < q < 2. Let f be the solution of (4.1)–(4.2). Then

there exists the limit

Lλ = limξ→∞ ξaf (ξ) ∈ (0, ∞).

Proof. We show as in [12] that the function v = f ′/f has a limit = 0 or = −∞ atξ = ∞. We claim that = 0.

Consider g(ξ) = ξa−1(f ′ + ξ2 f ). This auxiliary function satisfies the relation g′(ξ) =

ξa−1(λ|f ′|q + a−nξ

f ′) ≥ 0 because a < n and f ′ < 0. Then, given ξ0 > 0 there exists a

constant K > 0 such that f ′(ξ)+ ξ2 f (ξ) ≥ Kξ1−a for all ξ ≥ ξ0. In other words, we have

(f (ξ)eξ2/4)′ ≥ Kξ1−aeξ2/4, and hence

f (ξ) ≥ e−ξ2/4(

f (ξ0) + K

∫ ξ

ξ0

η1−aeη2/4 dη

).

Since limξ→∞(∫ ξ

ξ0η1−aeη2/4 dη)/(ξ−aeξ2/4) = 2, we have lim infξ→∞ ξaf (ξ) ≥ 2K > 0

which rules out the possibility that limξ→∞ f ′(ξ)/f (ξ) = −∞, because in such a casef (ξ) = O(e−kξ ) as ξ → ∞ for any k ∈ R.

The auxiliary function v = f ′/f satisfies the equation

v′(ξ) +(

1

2ξ + n − 1

ξ

)v(ξ) = −a

2+ ϕ(ξ),

where ϕ = λ|v|qf q−1 − v2, and limξ→∞ v(ξ) = 0. Therefore

v(ξ) =∫ ξ

0 τn−1eτ 2/4(− a2 + ϕ(τ)) dτ

ξn−1eξ2/4, ξ > 0,

and using the de l’Hopital rule we deduce as in [16, 11] that

f (ξ) = Lλξ−a(1 + o(1)).

�

Proof of Theorem 2.4. It is sufficient to prove the existence of the limit

B = limξ→∞ ξ2 (ξv(ξ) + a) .

In fact, by Lemma 4.2 and by an application of the de l’Hopital rule to the quotient(ξv + a)/ξ−2, we deduce that

B = B(λ) = 2λaqLq−1λ − 2a2 + 2(n − 2)a.

88 piotr biler, mohammed guedda and grzegorz karch J.evol.equ.

Consequently, we may write

f ′

f= −a

ξ+ B

ξ3+ o(ξ−3) as ξ → ∞.

A simple integration gives the following approximation of the solution f of the problem(4.1)–(4.2):

f (ξ) = Lλξ−a

{1 − B

2ξ2+ o

(1

ξ2

)}, as ξ → ∞.

Finally, since L(β) = βLβq−1 , Remark 4.1 on the scaling of solutions completes the proofof Theorem 2.4.

Proof of Proposition 2.10. Let β1 > β2 > 0 and fi be a solution of (2.7)–(2.8), whereβ = βi , i = 1, 2, respectively. The function h = f1 − f2 satisfies

h′′ + c(ξ)h′ = −a

2h − 1

2ξh′ − n − 1

ξh′, (4.7)

where

c(ξ) ={

−|f ′1|q (ξ)−|f ′

2|q (ξ)

f ′1(ξ)−f ′

2(ξ)if f ′

1(ξ) �= f ′2(ξ),

q|f ′1(ξ)|q−1 if f ′

1(ξ) = f ′2(ξ).

Note that 0 < c(ξ) → 0 as ξ → ∞, and the equation (4.7) can be written as

(eg(ξ)h′)′ = −a

2heg(ξ), (4.8)

where

g(ξ) = 1

4ξ2 +

∫ ξ

0c(s) ds.

First we prove that h > 0 and then h′ < 0 on (0, ∞). Actually, since h(0) > 0, there existsξ0 > 0 such that h(ξ) > 0 for all 0 ≤ ξ < ξ0. Suppose that h changes the sign and let ξ1

be the least zero of h. Integrating (4.7) over (0, ξ), ξ ≤ ξ1, yields

h′(ξ) = −a

2ξ1−ne−g(ξ)

∫ ξ

0h(s)s1−neg(s) ds.

Therefore, h′(ξ) < 0 for each 0 < ξ ≤ ξ1. On the other hand, we have

h′′ + c(ξ)h′ + 1

2(ξh)′ =

(1

2− a

2

)h − n − 1

ξh′,

Vol. 4, 2004 Viscous Hamilton-Jacobi equation 89

and an integration of the latter implies that h′(ξ1) is nonnegative, so a contradiction.Consequently h′(ξ) < 0 on (0, ∞), h is nonnegative, and h′, h → 0 as ξ → ∞. Arguingas in the proof of Theorem 2.4, we deduce that h satisfies

h(ξ) = Lξ−a (1 + o(1)) , L > 0.

Therefore L(β1) − L(β2) = L, and this implies that L(β1) > L(β2) completing the proof.

5. Weakly nonlinear large time behavior

This section is devoted to the proof of Theorem 2.11. However, first we need optimaldecay estimates of the Lp-norms of u and ∇u.

LEMMA 5.1. Assume that u0 satisfies (2.13) with b ∈ (0, n). For every p ∈ (n/b, ∞]there exists a constant C independent of t such that

‖u(·, t)‖p ≤ C(1 + t)−b/2+n/(2p) (5.1)

for all t > 0.

Proof. To prove these inequalities, observe that (1 + |x|)−b ∈ Lp(Rn) provided p ∈(n/b, ∞]. Hence, this lemma is an immediate consequence of the fact that et�u0 is a super-solution of the problem (1.1)–(1.2) (so that 0 ≤ u(x, t) ≤ et�u0(x)), and of the standardestimates of the heat semigroup (see (3.3)). �

PROPOSITION 5.2. Assume that u(x, t) is a solution of (1.1)–(1.2) with u0 satisfying(2.13). Let q and b satisfy (2.14). For every p ∈ (n/b, ∞] there exists a constant C suchthat

‖∇u(·, t)‖p ≤ C(1 + t)−b/2+n/(2p)t−1/2 (5.2)

for all t > 0.

Proof. Our proof begins with the observation that combining (3.5) with (5.1) we obtain

‖∇u(q−1)/q(·, t)‖∞ ≤ C(1 + t)−b(q−1)/(2q)t−1/2 (5.3)

for all t > 0 and a constant C.First, we prove (5.2) for p ∈ (nq/b, ∞]. Computing the Lp-norm of the gradient

∇u = qq−1u1/q∇u(q−1)/q , and using (5.1) (with p replaced by p/q) and (5.3) we have

‖∇u(·, t)‖p ≤ C‖u(·, t)‖1/qp/q‖∇u(q−1)/q(·, τ )‖∞

≤ C(1 + t)−b/2+n/(2p)t−1/2 (5.4)

for all t > 0 and C independent of t .

90 piotr biler, mohammed guedda and grzegorz karch J.evol.equ.

Next, let us consider p ∈ (n/b, nq/b] in (5.2). It follows immediately from (2.16) that

‖∇u(·, t)‖p ≤ ‖∇et�u0‖p +∫ t

0‖∇e(t−τ)�|∇u(·, τ )|q‖p dτ. (5.5)

We deal with the first term on the right hand side using standard properties of the heatsemigroup: for every p ∈ (n/b, ∞] we obtain

‖∇et�u0‖p = ‖∇e(t/2)�e(t/2)�u0‖p ≤ C(t/2)−1/2‖e(t/2)�u0‖p

≤ C(1 + t)−b/2+n/(2p)t−1/2

(cf. also Lemma 5.1).We split the integration range with respect to τ in the second term on the right hand side

of (5.5) into [0, t/2] ∪ [t/2, t], and we study each integral term separately. From now on,we use the inequality (5.2), already proved for each p ∈ (nq/b, ∞].

Standard estimates of the heat semigroup and (5.2) (note that qp > nq/b) give

∫ t

t/2‖∇e(t−τ)�|∇u(·, τ )|q‖p dτ ≤ C

∫ t

t/2(t − τ)−1/2‖∇u(·, τ )‖q

qp dτ

≤ C

∫ t

t/2(t − τ)−1/2(1 + τ)(−b/2+n/(2qp))qτ−q/2 dτ.

It is immediate that for t ∈ (0, 1] the last integral on the right hand side is bounded (recallthat q < 2) by

∫ t

t/2(t − τ)−1/2τ−q/2 dτ = Ct(1−q)/2 ≤ Ct−1/2.

On the other hand, for t ≥ 1 this integral can be estimated by

∫ t

t/2(t − τ)−1/2τ (−b/2+n/(2qp))q−q/2 dτ = Ct−bq/2+n/(2p)−q/2+1/2

≤ Ct−b/2+n/(2p)−1/2.

Here, we have used the inequality

−bq

2+ n

2p− q

2+ 1

2< −b

2+ n

2p− 1

2

which is equivalent to a = (2 − q)/(q − 1) < b.

Vol. 4, 2004 Viscous Hamilton-Jacobi equation 91

We proceed analogously in the case of the integral over [0, t/2] in (5.5) to prove thatthis is bounded by Ct−1/2 for t ∈ (0, 1].

Now, let us study the integral over [0, t/2] for t ≥ 1 more carefully. For everyn/b < s ≤ p we obtain

∫ t/2

0‖∇e(t−τ)�|∇u(·, τ )|q‖p dτ

≤ C

∫ t/2

0(t − τ)−n(1/s−1/p)/2−1/2‖∇u(·, τ )‖q

qs dτ

≤ C(t/2)−n(1/s−1/p)/2−1/2∫ t/2

0(1 + τ)(−b/2+n/(2qs))qτ−q/2 dτ. (5.6)

Here, we are allowed to use (5.2), because sq > nq/b.Observe now that (−b/2 + n/(2qs))q − q/2 < −1. Indeed, the left hand side of this

inequality is decreasing as a function of s. Moreover, for s = n/b, we obtain an inequalitywhich is equivalent to a = (2 − q)/(q − 1) < b. Hence the integral on the right hand sideof (5.6) is uniformly bounded with respect to t .

Next, note that the exponent in the first factor on the right hand side of (5.6) satisfies

−n

2

(1

s− 1

p

)− 1

2> −b

2+ n

2p− 1

2,

and the equality holds true if and only if s = n/b. These considerations allow us, however,to say that for every ε > 0 there is s ∈ (n/b, p) such that

‖∇u(·, t)‖p ≤ C(1 + t)−b/2+n/(2p)+εt−1/2 (5.7)

for each p ∈ (n/b, nq/b] and all t > 0. Now, our goal is to show that one can put ε = 0in (5.7). For this reason, we come back to inequality (5.6) and use (5.7) to obtain

∫ t/2

0‖∇e(t−τ)�|∇u(·, τ )|q‖p dτ

≤ C(t/2)−n(1/s−1/p)/2−1/2∫ t/2

0(1 + τ)(−b/2+n/(2qs)+ε)qτ−q/2 dτ.

Note that, by (5.7), we require qs > n/b, only. To conclude our reasoning, we chooses ∈ (n/(bq), n/b) (hence, t−n(1/s−1/p)/2−1/2 < t−b/2+n/(2p)−1/2 for t > 1) such that(−b/2+n/(2qs)+ε)q−q/2 < −1, which is always possible provided ε > 0 is sufficientlysmall. �

92 piotr biler, mohammed guedda and grzegorz karch J.evol.equ.

Proof of Theorem 2.11. The reasoning here is similar to that in the proof of Proposition 5.2.In view of the integral Equation (2.16), it suffices to find appropriate estimates of the Lp-norm of

∫ t

0 e(t−τ)�|∇u(τ)|q dτ . Applying standard estimates of the heat semigroup andthe estimates (5.2) for ∇u we obtain∫ t

t/2‖e(t−τ)�|∇u(·, τ )|q‖p dτ ≤ C

∫ t

t/2‖∇u(·, τ )‖q

qp dτ

≤ C

∫ t

t/2(1 + τ)(−b/2+n/(2qp)−1/2)q dτ

= Ct−b/2+n/(2p)+(q−1)(a−b)/2

for all t > 0 and a constant C. Since a < b, the right hand side tends to 0 as t → ∞ fasterthan t−b/2+n/(2p).

To deal with the integral over [0, t/2], we fix s < p (it will be precised later), and wecompute∫ t/2

0‖e(t−τ)�|∇u(·, τ )|q‖p dτ

≤ C

∫ t/2

0(t − τ)−n(1/s−1/p)/2‖∇u(·, τ )‖q

qs dτ

≤ C(t/2)−n(1/s−1/p)/2∫ t/2

0(1 + τ)(−b/2+n/(2qs))qτ−q/2 dτ.

Now, to get the conclusion we need to choose s as follows:

• s > n/(bq) (in order to be able to apply (5.2)),• s < n/b (in order to guarantee −n(1/s − 1/p)/2 < −b/2 + n/(2p)), and• s sufficiently close to n/b (to be sure that (−b/2 + n/(2qs))q − q/2 < −1).

Such a choice of s is possible, therefore this completes the proof of Theorem 2.11.

6. Integrable initial conditions

The theorem formulated below is based on ideas which are similar to those used alreadyin [21].

THEOREM 6.1. Assume that u = u(x, t) is the solution of the Cauchy problem for thelinear nonhomogeneous heat equation

ut = �u + f (x, t), x ∈ Rn, t > 0, (6.1)

u(x, 0) = u0(x), (6.2)

Vol. 4, 2004 Viscous Hamilton-Jacobi equation 93

with u0 ∈ L1(Rn) and f ∈ L1(Rn × (0, ∞)). Then

‖u(·, t) − M∞G(·, t)‖1 → 0 as t → ∞, (6.3)

where G is the heat kernel (1.5), and the constant

M∞ = limt→∞

∫R

nu(x, t) dx =

∫R

nu0(x) dx +

∫ ∞

0

∫R

nf (x, t) dx dt

is finite by the assumptions on u0 and f .

Proof. Obviously, the solution of (6.1)–(6.2) is of the form

u(t) = G(t) ∗ u0 +∫ t

0G(t − τ) ∗ f (τ) dτ.

Moreover, it is a well known property of solutions of the heat equation, cf. [14], that∥∥∥∥G(·, t) ∗ u0 −(∫

Rnu0(x) dx

)G(·, t)

∥∥∥∥1

→ 0 as t → ∞.

Hence, it suffices to show that∥∥∥∥∫ t

0G(·, t − τ) ∗ f (τ) dτ − G(·, t)

∫ ∞

0

∫R

nf (x, τ ) dx dτ

∥∥∥∥1

→ 0 (6.4)

as t → ∞. In order to prove the latter claim, note that we have∥∥∥∥G(·, t)∫ ∞

t

∫R

nf (x, τ ) dx dτ

∥∥∥∥1

≤∫ ∞

t

∫R

n|f (x, τ )| dx dτ → 0 as t → ∞.

This implies that the relation (6.4) holds true provided we shall show that

∫ t

0(G(·, t − τ) ∗ f (τ)) (x) dτ − G(x, t)

∫ t

0

∫R

nf (y, τ ) dy dτ

=∫ t

0

∫R

n(G(x − y, t − τ) − G(x, t)) f (y, τ ) dy dτ (6.5)

tends to 0 in L1(Rn) as t → ∞.To prove the latter assertion, we fix δ ∈ (0, 1) and decompose the integration range

Rn × [0, t] into two parts 1(t) and 2(t), where

1(t) = {y ∈ Rn : |y| ≤ δt1/2} × [0, δt],

2(t) = (Rn × [0, t])\ 1(t).

94 piotr biler, mohammed guedda and grzegorz karch J.evol.equ.

We estimate the L1-norm of the integral in (6.5) over 2(t) in a straightforward mannerusing properties of the heat kernel, by the following quantity∫∫

2(t)

(‖G(· − y, t − τ)‖1 + ‖G(·, t)‖1)|f (y, τ )| dy dτ

≤ 2∫∫

2(t)

|f (y, τ )| dy dτ.

The assumption of f guarantees that the right hand side tends to 0 as t → ∞.Next, using a change of variables and the selfsimilarity of G(x, t), we see that given

ε > 0 there exists δ > 0 such that

sup|y|≤δt1/20<τ≤δt

‖G(· − y, t − τ) − G(·, t)‖1 = sup|z|≤δ

0<s≤δ

‖G(· − z, 1 − s) − G(·, 1)‖1 ≤ ε.

As an immediate consequence of this estimate, we obtain that the L1-norm of the integralin (6.5) over 1(t) is bounded by ε

∫ ∞0

∫R

n |f (y, τ )| dy dτ .Since ε > 0 can be arbitrarily small, we conclude that (6.4) holds true and the proof of

Theorem 6.1 is complete. �

Proof of Theorem 2.14. The construction of the unique global in time solution for every(not necessarily nonnegative) initial datum from L∞(Rn) ∩ C(Rn) can be found in [15].By the maximum principle, this solution is nonnegative if u0 ≥ 0. If we integrate theEquation (1.1) with respect to x and t , we obtain after the integration by parts

∫R

nu(x, t) dx =

∫R

nu0(x) dx −

∫ t

0

∫R

n|∇u(x, τ )|q dx dτ. (6.6)

Hence, the limit M∞ = limt→∞∫

u(x, t) dx exists and is nonnegative. If fact, it is differentfrom zero, as it was shown in [10] and [3], provided q > (n+2)/(n+1). Finally, it followsfrom (6.6) that the function f (x, t) = −|∇u(x, t)|q is integrable over R

n × (0, ∞), hence(2.17) is an immediate consequence of Theorem 6.1.

Proof of Corollary 2.15. By the comparison principle, we have

0 ≤ u(x, t) ≤ G(·, t) ∗ u0(x)

for all (x, t) ∈ Rn × (0, ∞). Then, the standard estimates of the heat semigroup give

‖u(·, t)‖p ≤ ‖G(·, t) ∗ u0‖p ≤ Ct−n(1−1/p)/2‖u0‖1. (6.7)

Vol. 4, 2004 Viscous Hamilton-Jacobi equation 95

Finally, the proof of (2.18) is a direct consequence of (6.7) and the elementaryinequality

‖u(·, t) − M∞G(·, t)‖p ≤ C‖u(·, t) − M∞G(·, t)‖1/p

1

× (‖u(·, t)‖1−1/p∞ + ‖M∞G(·, t)‖1−1/p∞ )

valid for every p ∈ [1, ∞).An argument leading to the estimate of the L∞-norm is analogous to that used in (3.7)

provided we shall find an appropriate decay estimate of ‖∇u(·, t)‖p. Here, we use theinequality

‖∇u(q−1)/q(·, t)‖∞ ≤ (q − 1)1/2‖u0‖(q−1)/q

1 t−(q(n+1)−n)/(2q)

proved already in [3, Th.1]. Hence, computing the Lp-norm for p > q of the identity∇u = q

q−1u1/q∇u(q−1)/q we obtain

‖∇u(·, t)‖p ≤ q

q − 1‖u(·, t)‖1/q

p/q‖∇u(q−1)/q(·, t)‖∞ ≤ Ct−n(1−1/p)/2−1/2,

which completes the proof of Corollary 2.15.

Acknowledgements

This paper was written when the third author had a CNRS research fellowship atL.A.M.F.A., Universite de Picardie, Amiens. The first and the third author were partiallysupported by KBN grants 2/P03A/011/19, 50/P03/2000/18, and the second author-by DRI,UPJV (Amiens). M.G. and G.K. gratefully acknowledge the support of French-MoroccoScientific Cooperation Project “Action-Integree” No. 182/MA/99. Support by the Europeannetwork HYKE, founded by the EU (contract HPRN-CT-2002-00282), is acknowledged.The authors wish to express their gratitude to Philippe Laurencot for a valuable discussionand for sending them his papers concerning the Equation (1.1).

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Piotr Biler and Grzegorz KarchInstytut MatematycznyUniwersytet Wrocławskipl. Grunwaldzki 2/450–384 WrocławPolande-mail: piotr.biler@math.uni.wroc.pl

grzegorz.karch@math.uni.wroc.pl

Mohammed GueddaL.A.M.F.A./CNRS FRE 2270Universite de Picardie-Jules VerneFaculte de Mathematiques et d’Informatique33, rue Saint-Leu80039 AmiensFrancee-mail: mohamed.guedda@u-picardie.fr