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.