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Positive solutions of an elliptic equationwith a dynamical boundary condition
Marek Fila
Comenius University
Workshop in Nonlinear PDEs
Brussels, 2015
−∆u = f (u), x ∈ Ω ⊂ RN , t > 0,
∂tu + ∂νu = g(u), x ∈ ∂Ω, t > 0,
u(x ,0) = ϕ(x), x ∈ ∂Ω,
f ≡ 0, Ω – bounded
Lions (1969), Kirane (1992), F. and Quittner (1997), Yin (2003)
f ≡ 0, g(u) = up, p > 1, ϕ ≥ 0 and Ω = RN+
Amann and F. (1997), F., Ishige and Kawakami (2012)
f 6≡ 0, Ω – bounded
Escher (1992, 1994) – existence, uniqueness and smoothness
F. and Polácik (1999) – examples of local nonexistence andnonuniqueness if f is not globally Lipschitz
Gal and Meyries (2014) – blow-up, global existence, global attractorsif f (u) = h(u)− λu, h is globally Lipschitz and λ is large
f (u) = up, p > 1, g(u) ≡ 0, ϕ ≥ 0 and Ω = RN+
F., Ishige and Kawakami – this talk
−∆u = f (u), x ∈ Ω ⊂ RN , t > 0,
∂tu + ∂νu = g(u), x ∈ ∂Ω, t > 0,
u(x ,0) = ϕ(x), x ∈ ∂Ω,
f ≡ 0, Ω – bounded
Lions (1969), Kirane (1992), F. and Quittner (1997), Yin (2003)
f ≡ 0, g(u) = up, p > 1, ϕ ≥ 0 and Ω = RN+
Amann and F. (1997), F., Ishige and Kawakami (2012)
f 6≡ 0, Ω – bounded
Escher (1992, 1994) – existence, uniqueness and smoothness
F. and Polácik (1999) – examples of local nonexistence andnonuniqueness if f is not globally Lipschitz
Gal and Meyries (2014) – blow-up, global existence, global attractorsif f (u) = h(u)− λu, h is globally Lipschitz and λ is large
f (u) = up, p > 1, g(u) ≡ 0, ϕ ≥ 0 and Ω = RN+
F., Ishige and Kawakami – this talk
−∆u = f (u), x ∈ Ω ⊂ RN , t > 0,
∂tu + ∂νu = g(u), x ∈ ∂Ω, t > 0,
u(x ,0) = ϕ(x), x ∈ ∂Ω,
f ≡ 0, Ω – bounded
Lions (1969), Kirane (1992), F. and Quittner (1997), Yin (2003)
f ≡ 0, g(u) = up, p > 1, ϕ ≥ 0 and Ω = RN+
Amann and F. (1997), F., Ishige and Kawakami (2012)
f 6≡ 0, Ω – bounded
Escher (1992, 1994) – existence, uniqueness and smoothness
F. and Polácik (1999) – examples of local nonexistence andnonuniqueness if f is not globally Lipschitz
Gal and Meyries (2014) – blow-up, global existence, global attractorsif f (u) = h(u)− λu, h is globally Lipschitz and λ is large
f (u) = up, p > 1, g(u) ≡ 0, ϕ ≥ 0 and Ω = RN+
F., Ishige and Kawakami – this talk
−∆u = f (u), x ∈ Ω ⊂ RN , t > 0,
∂tu + ∂νu = g(u), x ∈ ∂Ω, t > 0,
u(x ,0) = ϕ(x), x ∈ ∂Ω,
f ≡ 0, Ω – bounded
Lions (1969), Kirane (1992), F. and Quittner (1997), Yin (2003)
f ≡ 0, g(u) = up, p > 1, ϕ ≥ 0 and Ω = RN+
Amann and F. (1997), F., Ishige and Kawakami (2012)
f 6≡ 0, Ω – bounded
Escher (1992, 1994) – existence, uniqueness and smoothness
F. and Polácik (1999) – examples of local nonexistence andnonuniqueness if f is not globally Lipschitz
Gal and Meyries (2014) – blow-up, global existence, global attractorsif f (u) = h(u)− λu, h is globally Lipschitz and λ is large
f (u) = up, p > 1, g(u) ≡ 0, ϕ ≥ 0 and Ω = RN+
F., Ishige and Kawakami – this talk
−∆u = f (u), x ∈ Ω ⊂ RN , t > 0,
∂tu + ∂νu = g(u), x ∈ ∂Ω, t > 0,
u(x ,0) = ϕ(x), x ∈ ∂Ω,
f ≡ 0, Ω – bounded
Lions (1969), Kirane (1992), F. and Quittner (1997), Yin (2003)
f ≡ 0, g(u) = up, p > 1, ϕ ≥ 0 and Ω = RN+
Amann and F. (1997), F., Ishige and Kawakami (2012)
f 6≡ 0, Ω – bounded
Escher (1992, 1994) – existence, uniqueness and smoothness
F. and Polácik (1999) – examples of local nonexistence andnonuniqueness if f is not globally Lipschitz
Gal and Meyries (2014) – blow-up, global existence, global attractorsif f (u) = h(u)− λu, h is globally Lipschitz and λ is large
f (u) = up, p > 1, g(u) ≡ 0, ϕ ≥ 0 and Ω = RN+
F., Ishige and Kawakami – this talk
The problem−∆u = f (u), x ∈ Ω ⊂ RN , t > 0,
∂tu + ∂νu = g(u), x ∈ ∂Ω, t > 0,
u(x ,0) = ϕ(x), x ∈ ∂Ω,
is the limit problem ofε∂tu −∆u = f (u), x ∈ Ω ⊂ RN , t > 0,
∂tu + ∂νu = g(u), x ∈ ∂Ω, t > 0,
u(x ,0) = φ(x), x ∈ Ω,
as ε→ 0, here ϕ = φ on ∂Ω.
The problem−∆u = f (u), x ∈ Ω ⊂ RN , t > 0,
∂tu + ∂νu = g(u), x ∈ ∂Ω, t > 0,
u(x ,0) = ϕ(x), x ∈ ∂Ω,
is the limit problem ofε∂tu −∆u = f (u), x ∈ Ω ⊂ RN , t > 0,
∂tu + ∂νu = g(u), x ∈ ∂Ω, t > 0,
u(x ,0) = φ(x), x ∈ Ω,
as ε→ 0, here ϕ = φ on ∂Ω.
−∆u = up, x ∈ RN
+, t > 0,
∂tu + ∂νu = 0, x ∈ ∂RN+, t > 0,
u(x ,0) = ϕ(x ′) ≥ 0, x = (x ′,0) ∈ ∂RN+,
where u = u(x , t), N ≥ 2, RN+ := x = (x ′, xN) : x ′ ∈ RN−1, xN > 0,
∆ is the N-dimensional Laplacian (in x), ∂ν := −∂/∂xN and p > 1.
write u = v + w where−∆v = (v + w)p, −∆w = 0 x ∈ RN
+, t > 0,
v = 0, ∂tw + ∂νw = −∂νv , x ∈ ∂RN+, t > 0,
w(x ,0) = ϕ(x ′) ≥ 0, x = (x ′,0) ∈ ∂RN+,
−∆u = up, x ∈ RN
+, t > 0,
∂tu + ∂νu = 0, x ∈ ∂RN+, t > 0,
u(x ,0) = ϕ(x ′) ≥ 0, x = (x ′,0) ∈ ∂RN+,
where u = u(x , t), N ≥ 2, RN+ := x = (x ′, xN) : x ′ ∈ RN−1, xN > 0,
∆ is the N-dimensional Laplacian (in x), ∂ν := −∂/∂xN and p > 1.
write u = v + w where−∆v = (v + w)p, −∆w = 0 x ∈ RN
+, t > 0,
v = 0, ∂tw + ∂νw = −∂νv , x ∈ ∂RN+, t > 0,
w(x ,0) = ϕ(x ′) ≥ 0, x = (x ′,0) ∈ ∂RN+,
−∆u = up, x ∈ RN
+, t > 0,
∂tu + ∂νu = 0, x ∈ ∂RN+, t > 0,
u(x ,0) = ϕ(x ′) ≥ 0, x = (x ′,0) ∈ ∂RN+,
where u = u(x , t), N ≥ 2, RN+ := x = (x ′, xN) : x ′ ∈ RN−1, xN > 0,
∆ is the N-dimensional Laplacian (in x), ∂ν := −∂/∂xN and p > 1.
write u = v + w where−∆v = (v + w)p, −∆w = 0 x ∈ RN
+, t > 0,
v = 0, ∂tw + ∂νw = −∂νv , x ∈ ∂RN+, t > 0,
w(x ,0) = ϕ(x ′) ≥ 0, x = (x ′,0) ∈ ∂RN+,
For any x ′ ∈ RN−1 and λ > 0, let P be the (N − 1)-dimensionalPoisson kernel
P(x ′, λ) := cNλ1−NΛ−N(x ′/λ) with Λ(x ′) := (1 + |x ′|2)1/2,
and
P(x ′, xN , t) := P(x ′, xN + t), (x ′, xN , t) ∈ RN+ × (0,∞).
G(x , y) :=
kN
(|x − y |−(N−2) − |x − y∗|−(N−2)
)if N ≥ 3,
14π
log(
1 +4x2y2
|x − y |2
)if N = 2,
G is the Green function for the Laplace equation on RN+ with the
Dirichlet boundary condition.
For any x ′ ∈ RN−1 and λ > 0, let P be the (N − 1)-dimensionalPoisson kernel
P(x ′, λ) := cNλ1−NΛ−N(x ′/λ) with Λ(x ′) := (1 + |x ′|2)1/2,
and
P(x ′, xN , t) := P(x ′, xN + t), (x ′, xN , t) ∈ RN+ × (0,∞).
G(x , y) :=
kN
(|x − y |−(N−2) − |x − y∗|−(N−2)
)if N ≥ 3,
14π
log(
1 +4x2y2
|x − y |2
)if N = 2,
G is the Green function for the Laplace equation on RN+ with the
Dirichlet boundary condition.
For any x ′ ∈ RN−1 and λ > 0, let P be the (N − 1)-dimensionalPoisson kernel
P(x ′, λ) := cNλ1−NΛ−N(x ′/λ) with Λ(x ′) := (1 + |x ′|2)1/2,
and
P(x ′, xN , t) := P(x ′, xN + t), (x ′, xN , t) ∈ RN+ × (0,∞).
G(x , y) :=
kN
(|x − y |−(N−2) − |x − y∗|−(N−2)
)if N ≥ 3,
14π
log(
1 +4x2y2
|x − y |2
)if N = 2,
G is the Green function for the Laplace equation on RN+ with the
Dirichlet boundary condition.
DefinitionLet ϕ be a nonnegative measurable function in RN−1. For any σ > 0,we call a nonnegative measurable function u in RN
+ × (0, σ] a solutionin RN
+ × (0, σ] if u satisfies
u(x ′, xN , t) =
∫RN−1
P(x ′ − y ′, xN , t)ϕ(y ′) dy ′ +
∫RN
+
G(x , y)u(y , t)p dy
+
∫ t
0
∫RN
+
P(x ′ − y ′, xN + yN , t − s)u(y , s)p dy ds <∞
for almost all x ′ ∈ RN−1 and all xN ∈ [0,∞) and t ∈ (0, σ].
Nonuniqueness
If v is a solution of−∆v = vp, x ∈ RN
+,
∂νv = 0, x ∈ ∂RN+,
v(x) = ϕ(x ′) ≥ 0, x = (x ′,0) ∈ ∂RN+,
thenu(x , t) := v(x ′, xN + t)
satisfies −∆u = up, x ∈ RN
+, t > 0,
∂tu + ∂νu = 0, x ∈ ∂RN+, t > 0,
u(x ,0) = ϕ(x ′) ≥ 0, x = (x ′,0) ∈ ∂RN+.
In fact, u is the minimal solution. Minimal solutions are unique.
Nonuniqueness
If v is a solution of−∆v = vp, x ∈ RN
+,
∂νv = 0, x ∈ ∂RN+,
v(x) = ϕ(x ′) ≥ 0, x = (x ′,0) ∈ ∂RN+,
thenu(x , t) := v(x ′, xN + t)
satisfies −∆u = up, x ∈ RN
+, t > 0,
∂tu + ∂νu = 0, x ∈ ∂RN+, t > 0,
u(x ,0) = ϕ(x ′) ≥ 0, x = (x ′,0) ∈ ∂RN+.
In fact, u is the minimal solution. Minimal solutions are unique.
Nonuniqueness
If v is a solution of−∆v = vp, x ∈ RN
+,
∂νv = 0, x ∈ ∂RN+,
v(x) = ϕ(x ′) ≥ 0, x = (x ′,0) ∈ ∂RN+,
thenu(x , t) := v(x ′, xN + t)
satisfies −∆u = up, x ∈ RN
+, t > 0,
∂tu + ∂νu = 0, x ∈ ∂RN+, t > 0,
u(x ,0) = ϕ(x ′) ≥ 0, x = (x ′,0) ∈ ∂RN+.
In fact, u is the minimal solution.
Minimal solutions are unique.
Nonuniqueness
If v is a solution of−∆v = vp, x ∈ RN
+,
∂νv = 0, x ∈ ∂RN+,
v(x) = ϕ(x ′) ≥ 0, x = (x ′,0) ∈ ∂RN+,
thenu(x , t) := v(x ′, xN + t)
satisfies −∆u = up, x ∈ RN
+, t > 0,
∂tu + ∂νu = 0, x ∈ ∂RN+, t > 0,
u(x ,0) = ϕ(x ′) ≥ 0, x = (x ′,0) ∈ ∂RN+.
In fact, u is the minimal solution. Minimal solutions are unique.
Theorem 1
(i) If
1 < p ≤ p∗ :=N + 1N − 1
,
then there are no nontrivial local-in-time solutions;
(ii) If p > p∗, then, for suitable small initial data ϕ, there existsolutions defined for all t > 0 which behave like the Poissonkernel as t →∞.
Theorem 1
(i) If
1 < p ≤ p∗ :=N + 1N − 1
,
then there are no nontrivial local-in-time solutions;
(ii) If p > p∗, then, for suitable small initial data ϕ, there existsolutions defined for all t > 0 which behave like the Poissonkernel as t →∞.
Theorem 2
Let p > p∗ and let ψ be a nonnegative function in RN−1 such that
lim inf|x ′|→∞
|x ′|2
p−1ψ(x ′) > 0.
Then there exists a constant κ > 0 such that, if
ϕ(x ′) ≥ κψ(x ′), x ′ ∈ RN−1,
then there is no local-in-time solution.
Theorem 3
Let p > p∗ and let ϕ be a nonnegative function in RN−1 such that
lim inf|x ′|→∞
|x ′|2
p−1ϕ(x ′) =∞.
Then there is no local-in-time solution.
Theorem 2
Let p > p∗ and let ψ be a nonnegative function in RN−1 such that
lim inf|x ′|→∞
|x ′|2
p−1ψ(x ′) > 0.
Then there exists a constant κ > 0 such that, if
ϕ(x ′) ≥ κψ(x ′), x ′ ∈ RN−1,
then there is no local-in-time solution.
Theorem 3
Let p > p∗ and let ϕ be a nonnegative function in RN−1 such that
lim inf|x ′|→∞
|x ′|2
p−1ϕ(x ′) =∞.
Then there is no local-in-time solution.
Theorem 2
Let p > p∗ and let ψ be a nonnegative function in RN−1 such that
lim inf|x ′|→∞
|x ′|2
p−1ψ(x ′) > 0.
Then there exists a constant κ > 0 such that, if
ϕ(x ′) ≥ κψ(x ′), x ′ ∈ RN−1,
then there is no local-in-time solution.
Theorem 3
Let p > p∗ and let ϕ be a nonnegative function in RN−1 such that
lim inf|x ′|→∞
|x ′|2
p−1ϕ(x ′) =∞.
Then there is no local-in-time solution.
Theorem 4
Let p > p∗. Then there exists k > 0 such that if
ϕ(x ′) ≤ k(1 + |x ′|)−2
p−1 , x ′ ∈ RN−1,
then there is a global-in-time solution u satisfying
u(x , t) ≤ C(1 + |x ′|+ xN + t)−2
p−1 , (x , t) ∈ RN+ × (0,∞),
for some constant C > 0.
(D)
−∆u = f (u), x ∈ RN
+, t > 0,∂tu + ∂νu = 0, x ∈ ∂RN
+, t > 0,u(x ,0) = ϕ(x ′), x = (x ′,0) ∈ ∂RN
+,
f is nondecreasing, continuous, f (0) ≥ 0.
Theorem 5 Let ϕ be a nonnegative measurable function in RN−1.Then the following statements are equivalent:
(a) There is a local-in-time solution;
(b) There is a global-in-time solution;
(c) There is a solution of
(E)
−∆v = f (v), x ∈ RN
+,
v(x) = ϕ(x ′), x = (x ′,0) ∈ ∂RN+.
Furthermore, if u = u(x , t) is a minimal solution of (D) and v = v(x) isa minimal solution of (E) then
u(x ′, xN , t) = v(x ′, xN + t)
for almost all x ′ ∈ RN−1 and all xN ≥ 0 and t > 0.
(D)
−∆u = f (u), x ∈ RN
+, t > 0,∂tu + ∂νu = 0, x ∈ ∂RN
+, t > 0,u(x ,0) = ϕ(x ′), x = (x ′,0) ∈ ∂RN
+,
f is nondecreasing, continuous, f (0) ≥ 0.
Theorem 5 Let ϕ be a nonnegative measurable function in RN−1.Then the following statements are equivalent:
(a) There is a local-in-time solution;
(b) There is a global-in-time solution;
(c) There is a solution of
(E)
−∆v = f (v), x ∈ RN
+,
v(x) = ϕ(x ′), x = (x ′,0) ∈ ∂RN+.
Furthermore, if u = u(x , t) is a minimal solution of (D) and v = v(x) isa minimal solution of (E) then
u(x ′, xN , t) = v(x ′, xN + t)
for almost all x ′ ∈ RN−1 and all xN ≥ 0 and t > 0.
(D)
−∆u = f (u), x ∈ RN
+, t > 0,∂tu + ∂νu = 0, x ∈ ∂RN
+, t > 0,u(x ,0) = ϕ(x ′), x = (x ′,0) ∈ ∂RN
+,
f is nondecreasing, continuous, f (0) ≥ 0.
Theorem 5 Let ϕ be a nonnegative measurable function in RN−1.Then the following statements are equivalent:
(a) There is a local-in-time solution;
(b) There is a global-in-time solution;
(c) There is a solution of
(E)
−∆v = f (v), x ∈ RN
+,
v(x) = ϕ(x ′), x = (x ′,0) ∈ ∂RN+.
Furthermore, if u = u(x , t) is a minimal solution of (D) and v = v(x) isa minimal solution of (E) then
u(x ′, xN , t) = v(x ′, xN + t)
for almost all x ′ ∈ RN−1 and all xN ≥ 0 and t > 0.
(D)
−∆u = f (u), x ∈ RN
+, t > 0,∂tu + ∂νu = 0, x ∈ ∂RN
+, t > 0,u(x ,0) = ϕ(x ′), x = (x ′,0) ∈ ∂RN
+,
f is nondecreasing, continuous, f (0) ≥ 0.
Theorem 5 Let ϕ be a nonnegative measurable function in RN−1.Then the following statements are equivalent:
(a) There is a local-in-time solution;
(b) There is a global-in-time solution;
(c) There is a solution of
(E)
−∆v = f (v), x ∈ RN
+,
v(x) = ϕ(x ′), x = (x ′,0) ∈ ∂RN+.
Furthermore, if u = u(x , t) is a minimal solution of (D) and v = v(x) isa minimal solution of (E) then
u(x ′, xN , t) = v(x ′, xN + t)
for almost all x ′ ∈ RN−1 and all xN ≥ 0 and t > 0.
(D)
−∆u = f (u), x ∈ RN
+, t > 0,∂tu + ∂νu = 0, x ∈ ∂RN
+, t > 0,u(x ,0) = ϕ(x ′), x = (x ′,0) ∈ ∂RN
+,
f is nondecreasing, continuous, f (0) ≥ 0.
Theorem 5 Let ϕ be a nonnegative measurable function in RN−1.Then the following statements are equivalent:
(a) There is a local-in-time solution;
(b) There is a global-in-time solution;
(c) There is a solution of
(E)
−∆v = f (v), x ∈ RN
+,
v(x) = ϕ(x ′), x = (x ′,0) ∈ ∂RN+.
Furthermore, if u = u(x , t) is a minimal solution of (D) and v = v(x) isa minimal solution of (E) then
u(x ′, xN , t) = v(x ′, xN + t)
for almost all x ′ ∈ RN−1 and all xN ≥ 0 and t > 0.
(D)
−∆u = f (u), x ∈ RN
+, t > 0,∂tu + ∂νu = 0, x ∈ ∂RN
+, t > 0,u(x ,0) = ϕ(x ′), x = (x ′,0) ∈ ∂RN
+,
f is nondecreasing, continuous, f (0) ≥ 0.
Theorem 5 Let ϕ be a nonnegative measurable function in RN−1.Then the following statements are equivalent:
(a) There is a local-in-time solution;
(b) There is a global-in-time solution;
(c) There is a solution of
(E)
−∆v = f (v), x ∈ RN
+,
v(x) = ϕ(x ′), x = (x ′,0) ∈ ∂RN+.
Furthermore, if u = u(x , t) is a minimal solution of (D) and v = v(x) isa minimal solution of (E) then
u(x ′, xN , t) = v(x ′, xN + t)
for almost all x ′ ∈ RN−1 and all xN ≥ 0 and t > 0.
Theorem 6
Let p∗ < p <∞ and consider the elliptic problem−∆v = vp in RN
+,
v = κψ on ∂RN+,
where κ > 0 and ψ = ψ(x ′) is a nonnegative bounded function inRN−1 such that
ψ 6≡ 0 in RN−1 and lim sup|x ′|→∞
|x ′|2
p−1ψ(x ′) <∞.
Then there exists κ∗ ∈ (0,∞) such that
I If 0 < κ < κ∗ then there is a solution.
I If κ > κ∗ then there is no solution.
Theorem 6
Let p∗ < p <∞ and consider the elliptic problem−∆v = vp in RN
+,
v = κψ on ∂RN+,
where κ > 0 and ψ = ψ(x ′) is a nonnegative bounded function inRN−1 such that
ψ 6≡ 0 in RN−1 and lim sup|x ′|→∞
|x ′|2
p−1ψ(x ′) <∞.
Then there exists κ∗ ∈ (0,∞) such that
I If 0 < κ < κ∗ then there is a solution.
I If κ > κ∗ then there is no solution.
Theorem 6
Let p∗ < p <∞ and consider the elliptic problem−∆v = vp in RN
+,
v = κψ on ∂RN+,
where κ > 0 and ψ = ψ(x ′) is a nonnegative bounded function inRN−1 such that
ψ 6≡ 0 in RN−1 and lim sup|x ′|→∞
|x ′|2
p−1ψ(x ′) <∞.
Then there exists κ∗ ∈ (0,∞) such that
I If 0 < κ < κ∗ then there is a solution.
I If κ > κ∗ then there is no solution.
proofs – integral representation + Phragmén-Lindelöf
Theorem 7
Let σ > 0 and let u = u(x , t) satisfy
u(·, t) ∈ C2(RN+) ∩ C1(RN
+) for any t ∈ (0, σ],
u ∈ C(RN+ × (0, σ]), ∂tu ∈ C(∂RN
+ × (0, σ]),
and
−∆u ≥ 0 in RN+ × (0, σ], ∂tu + ∂νu ≥ 0 on RN
+ × (0, σ].
Assume that
lim inft→+0
infx ′∈B′(0,R)
u(x ′,0, t) ≥ 0 for any R > 0,
lim supR→∞
inf|x|=R,t∈(0,σ]
u(x , t)1 + xN
≥ 0.
Then u ≥ 0 in RN+ × (0, σ].
proofs – integral representation + Phragmén-Lindelöf
Theorem 7
Let σ > 0 and let u = u(x , t) satisfy
u(·, t) ∈ C2(RN+) ∩ C1(RN
+) for any t ∈ (0, σ],
u ∈ C(RN+ × (0, σ]), ∂tu ∈ C(∂RN
+ × (0, σ]),
and
−∆u ≥ 0 in RN+ × (0, σ], ∂tu + ∂νu ≥ 0 on RN
+ × (0, σ].
Assume that
lim inft→+0
infx ′∈B′(0,R)
u(x ′,0, t) ≥ 0 for any R > 0,
lim supR→∞
inf|x|=R,t∈(0,σ]
u(x , t)1 + xN
≥ 0.
Then u ≥ 0 in RN+ × (0, σ].
proofs – integral representation + Phragmén-Lindelöf
Theorem 7
Let σ > 0 and let u = u(x , t) satisfy
u(·, t) ∈ C2(RN+) ∩ C1(RN
+) for any t ∈ (0, σ],
u ∈ C(RN+ × (0, σ]), ∂tu ∈ C(∂RN
+ × (0, σ]),
and
−∆u ≥ 0 in RN+ × (0, σ], ∂tu + ∂νu ≥ 0 on RN
+ × (0, σ].
Assume that
lim inft→+0
infx ′∈B′(0,R)
u(x ′,0, t) ≥ 0 for any R > 0,
lim supR→∞
inf|x|=R,t∈(0,σ]
u(x , t)1 + xN
≥ 0.
Then u ≥ 0 in RN+ × (0, σ].
proofs – integral representation + Phragmén-Lindelöf
Theorem 7
Let σ > 0 and let u = u(x , t) satisfy
u(·, t) ∈ C2(RN+) ∩ C1(RN
+) for any t ∈ (0, σ],
u ∈ C(RN+ × (0, σ]), ∂tu ∈ C(∂RN
+ × (0, σ]),
and
−∆u ≥ 0 in RN+ × (0, σ], ∂tu + ∂νu ≥ 0 on RN
+ × (0, σ].
Assume that
lim inft→+0
infx ′∈B′(0,R)
u(x ′,0, t) ≥ 0 for any R > 0,
lim supR→∞
inf|x|=R,t∈(0,σ]
u(x , t)1 + xN
≥ 0.
Then u ≥ 0 in RN+ × (0, σ].
Thanks for your attention.
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