Global in time existence of solutions for the heat equation with a superlinear source term

Global in time existence of solutions for the heat equation with a superlinear

source term

Yohei Fujishima (Shizuoka Univ.) joint work with Prof. Norisuke Ioku (Ehime Univ.)

Workshop on Nonlinear Parabolic PDEs Institut Mittag-Leffler, 11–15 June. 2018

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Introduction

Consider (P)

{ ∂_tu = ∆u + f (u), x ∈ R^N, t > 0, u(x , 0) = u₀(x ) ≥ 0, x ∈ R^N.

f ∈ C¹([0,∞)): non-negative, increasing function.

Aim: Global in time existence of sol. of problem (P).

Definition

For a suitable Banach space X (e.g. X = L^r(R^N) or L^r_ul(R^N)), u = u(x , t)∈ C^2,1(R^N× (0, T )) is a classical sol. of (P) in X if

• u satisfies the equation pointwisely.

• lim

t→0∥u(·, t) − e^t∆u₀∥X = 0. (not lim

t→0∥u(·, t) − u0∥X = 0) (e^t∆u0)(x ) = (4πt)^−N2

∫

R^N

e^{−|x−y|24t} u0(y ) dy : sol. of the heat eq.

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Existence and Non-existence of solutions





Case u₀ ∈ L^∞(R^N)⇝ ∀u0 ∈ L^∞,∃ classical sol. of (P) in L^∞.







Case u0 ̸∈ L^∞(R^N)⇝ Singularity vs. Nonlinearity Weissler ’80: Let u₀ ∈ L^r(R^N) (r ≥ 1), f (u) = u^p (p > 1).

Putr_c := ^N₂(p− 1).

• r > r_c or r = r_c > 1 ⇒ ∀u0 ∈ L^r, ∃ classical sol. of (P) in L^r.

• 1≤ r < rc ⇒ ∃u0(≥ 0) ∈ L^r s.t. ̸ ∃ nonnegative sol. in L^r. Critical space

L^rc(R^N) classifies the existence and nonexistence.

Remark: Letu_λ(x , t) := λ^p−12 u(λx , λ²t) for λ > 0 (Self-similar scaling). If u satisfies ∂_tu = ∆u + u^p, then so does u_λ for all λ > 0.

∥uλ(·, 0)∥L^r =∥u0∥L^r ⇐⇒ r = N

2(p− 1)(= r_c).

⇝ L^rc(R^N) is the scale invariant space.

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Existence and Non-existence of solutions

F.-Ioku, to appear:

Definition (Uniformly local L^p space, p ≥ 1) L^p_ul(R^N) :=

{

u ∈ L^p_loc : ∥u∥p,ul := sup

y∈R^N

(∫

B1(y )

|u(x)|^pdx )¹_p

<∞ }

, L^p(R^N)⊊ L^p_ul(R^N) := BUC (R^N)^∥·∥p,ul ⊊ L^p_ul(R^N) (1≤ p < ∞).

Let F (s) :=

∫ _∞

s

1

f (u)du < ∞ (s > 0) and assume that the limit A exists:

A := lim

s→∞f^′(s)F (s) =⇒ A ≥ 1.

Putr_c(A) := ^N₂ ·_A¹₋₁ (A > 1).

Remark: f (u) = u^p ⇒ f^′(s)F (s) = _p−1^p (= A), r_c(A) = ^N₂(p− 1).

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Existence and Non-existence of solutions

Assume f^′(s)F (s)≤ A for s >> 1.

Existence:

• (A > 1) r > r_c(A) or r = r_c(A) > 1

⇒ ∀u0 with 1

F (u₀)^A−1 ∈ L^rul,∃ classical sol. of (P) in L^rul.

• (A = 1) r ≥ ^N₂ ⇒ ∀u0 with 1

F (u₀)^r ∈ L¹ul, ∃ sol. of (P) in L^∞. Nonexistence: 1≤ r < rc(A) (A > 1) or 0 < r < ^N₂ (A = 1)

⇒ ∃u0 ≥ 0 s.t. ̸ ∃ nonnegative sol. of (P) in L^r_ul or L^∞.

Remark: Letu_λ(x , t) := F⁻¹[λ⁻²F (u(λx , λ²t))] for λ > 0 (Quasi scaling). If u satisfies ∂_tu = ∆u + f (u), then

∂_tu_λ = ∆u_λ+ f (u_λ) + |∇uλ|²

f (u_λ)F (u_λ)[f^′(u)F (u)− f^′(u_λ)F (u_λ)].

On the other hand,

∫

R^N

1

F (u_λ(x , 0))^N2 dx =

∫

R^N

1

F (u₀(x ))^N2 dx.

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Remarks

• 1

F (s)^N2 → ∞ as s → ∞ ⇝ { 1

F (u₀(x ))^N2 =∞}

={u0 =∞}

• Let A > 1 and f^′(s)F (s)≤ A (s >> 1). Then s ≲ _{F (s)}¹A−1. This implies that u₀ ∈ L^rul proveded that 1

F (u₀)^A−1 ∈ L^rul.

⇝ lim

t→0∥u(t) − e^t∆u₀∥L^r_ul = 0 (expected singularity: L^r_ul)

• f (u) = u^p ⇒ F (s) = _p−1¹ s^−(p−1), f^′(s)F (s) = _p−1^p = A > 1.

1

F (u₀)^A−1 ∈ L^rul ⇐⇒ u0 ∈ L^rul, r_c(A) = N

2(p− 1).

• It follows that





f (u) = u^p ⇒ A = _p−1^p → 1 (p → ∞), f (u) = e^u ⇒ A = 1,

f (u) = e^u2 ⇒ A = 1.

Furthermore, f^′(s)F (s)≤ 1 if f (u) = e^u or e^u2 (s >> 1).

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Main Theorem

Assumef (0) = 0 and the existence ofα := lim

s→0f^′(s)F (s) ⇒ α ≥ 1.

Theorem (Global existence)

Assume that f^′(s)F (s)≤ A (s >> 1), f^′(s)F (s)≤ α (0 < s << 1), f^′(s)F (s)≤ max{α, A} (s > 0), α < 1 + ^N₂. If

∫

R^N

1

F (u₀)^N2 dx << 1,

then ∃ global in time sol. of (P) in L^rul^c(A) (A > 1) or L^∞ (A = 1).

Remark: (1)

∫

R^N

1

F (u_λ(x , 0))^N2 dx =

∫

R^N

1

F (u₀(x ))^N2 dx (λ > 0) (2) f (u) = u^p ⇒ α = _p−1^p .

α < 1 + N

2 ⇐⇒ p > 1 + 2

N (Fujita exponent)

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Idea of the proof

Local in time existence for the case A > 1 and r = r_c(A) We construct asuper-solution u of (P), that is,

u(x , t)≥ (e^t∆u₀)(x ) +

∫ _t

0

[e^(t−s)∆f (u(s))](x ) ds.

⇝ ∃ sol. u of (P) satisfying0≤ u(x, t) ≤ u(x, t). In fact, define {un}^∞n=0 by u₀(x , t) = (e^t∆u₀)(x ) and

u_n+1(x , t) = (e^t∆u₀)(x ) +

∫ t 0

[e^(t−s)∆f (u_n(s))](x ) ds.

Then 0≤ un(x , t)≤ un+1(x , t)≤ u(x, t) for all n = 0, 1, ....

⇝u(x , t) := lim

n→∞u_n(x , t) gives a sol. of (P).

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Idea of the proof

Let v be the sol. of

∂_tv = ∆v + (A− 1)v^A−1A , v (0) = 1

F (u₀)^A−1 ∈ L^r_ul^c(A).

↑

u₀ = max{u0(x ), k}(k >> 1)

Note that r_c(A) = ^N₂ ·( _A

A−1 − 1)

. ⇝ ∃v: sol.

Put u(x , t) := F⁻¹(v (x , t)^−A−11 ).

⇝ u(x, t) ≥ k

=⇒ ∂tu = ∆u + f (u) + |∇u|²

f (u)F (u)[A− f^′(u)F (u)]

| {z }

≥0 (u>>1)

.

u: supersolution ⇝ ∃u: sol. of (P) s.t. u(x, t) ≤ u(x, t). Remark

inf

x∈R^Nu(x , 0) > 0 =⇒ u blows up in finite time

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Idea of the proof

Let v be the sol. of

∂_tv = ∆v + (A− 1)v^A−1A , v (0) = 1

F (u₀)^A−1 ∈ L^r_ul^c(A).

↑

u₀ = max{u0(x ), k}(k >> 1) Note that r_c(A) = ^N₂ ·( _A

A−1 − 1)

. ⇝ ∃v: sol.

Put u(x , t) := F⁻¹(v (x , t)^−A−11 ). ⇝ u(x, t) ≥ k

=⇒ ∂tu = ∆u + f (u) + |∇u|²

f (u)F (u)[A− f^′(u)F (u)]

| {z }

≥0 (u>>1)

.

u: supersolution ⇝ ∃u: sol. of (P) s.t. u(x, t) ≤ u(x, t).

Remark inf

x∈R^Nu(x , 0) > 0 =⇒ u blows up in finite time

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Global existence for the case α > A > 1

Assume α > A and f^′(s)F (s)≤ α for all s > 0.

We consider the following equations:

∂_tu = ∆u + f (u), (P)

∂tv = ∆v + (A− 1)v^A−1A , (PA)

↑

local in time existence

∂_tw = ∆w + (α− 1)w^αα−1. (P_α)

s f^′(s)F (s)

A α

1

1st step: w : sol. of (P_α) with w (0) = 1

F (u₀)^α−1. Then v (x , t) := w (x , t)^α−1A−1 =⇒ ∂tv ≥ ∆v + (A − 1)v^A−1A .

⇝ ∃v: sol. of (PA) with v (0) = 1

F (u₀)^α−1 s.t. v ≤ v = w^Aα−1−1.

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Global existence for the case α > A > 1

2nd step (Local existence):

• u(x , t) := F⁻¹(v (x , t)^−A−11 )⇝ u is a supersolution of (P)

• ∃u: sol. of (P) with u(0) = u0 satisfying u ≤ u ≤ F⁻¹(w^−α−11 ) 3rd step (Global existence):

w (x , t) := 1

F (u(x , t))^α−1 =⇒ ∂tw ≤ ∆w + (α − 1)w^αα−1. Define{Wn}^∞n=0 by W₀(x , t) = w (x , t) ≤ w(x, t)and

W_n+1(x , t) := e^t∆

( 1

F (u₀(x ))^α−1 )

+ (α− 1)

∫ _t

0

e^(t−s)∆W_n(s)^αα−1 ds.

Thenw = F (u)^−(α−1)≤ Wn≤ Wn+1≤ w.

⇝ W (x, t) := lim

n→∞Wn(x , t): sol. of (Pα) s.t. F (u)^−(α−1) ≤ W .

⇐⇒ u(x , t)≤ F⁻¹(W (x , t)^−α−11 )

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Global existence for the case α > A > 1

Weissler ’81

Letrc = ^N₂(p− 1) > 1 and u0 ∈ L^rc(R^N). If∥u0∥L^rc is sufficiently small, then ∃ global solution u of

∂_tu = ∆u + u^p, u(0) = u₀. Since

W (·, 0) = 1

F (u₀)^α−1 ∈ L^N2·α−11 = L^{N2(α−1α −1)}, 1

F (u₀)^α−1

N 2·α−1¹

L

N 2 · 1

α−1

=

∫

R^N

1

F (u₀(x ))^N2 dx : small, W : sol. of (Pα), exists globally in time.

(P_α) ∂_tw = ∆w + (α− 1)w^αα−1

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Applications

(P1) ∂_tu = ∆u + u^p+ u^q, x ∈ R^N, t > 0. (p > q > 1)

• f^′(s)F (s)≤ _p−1^p = lim

s→∞f^′(s)F (s) = A for s >> 1.

• f^′(s)F (s) < _q−1^q = lim

s→0f^′(s)F (s) = α for all s > 0.

• F (s)^−N2 ≲ s^N2(p−1)+ s^N2(q−1) for all s > 0. (F (s) =∫_∞

s du f (u)) Corollary 1

Assume q > 1 + ^N₂ (⇐⇒ α < 1 + ^N₂). If

∫

R^N

(|u0(x )|^N2(p−1)+|u0(x )|^N2(q−1)) dx << 1, then ∃u: global sol. of (P1) satisfying

tlim→0∥u(·, t) − u0∥

L

N 2(p−1) ul

= 0, lim

t→0∥u(·, t) − u0∥_L^N₂(q−1) = 0.

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Applications

Let f (u) = e^u2 and consider

(P2) ∂_tu = ∆u + e^u2, x ∈ R^N, t > 0.

• f^′(s)F (s) < 1 = lim

s→∞f^′(s)F (s) = A for all s > 0.

•

∫

R^N|u0|^N2e^N2|u0|2dx < ∞ ⇒

∫

R^N

1

F (u₀(x ))^N2 dx <∞ (thereshold integrability for local existence)

⇝ ∃u: local sol. of (P2) s.t. lim

t→0∥u(t) − e^t∆u₀∥∞ = 0 and

tlim→0 sup

y∈R^N

∫

B1(y )

|u(t) − u0|^N2e^{N2|u(t)−u0|2}dx = 0.

Remark: Since f (0) > 0 and u₀(x ) ≥ 0,u blows up in finite time.

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Applications

Let f (u) = u^λe^u2 (λ > 1 + _N²) and consider

(P3) ∂_tu = ∆u + u^λe^u2, x ∈ R^N, t > 0.

• f^′(s)F (s) < 1 = lim

s→∞f^′(s)F (s) = A for s >> 1

• f^′(s)F (s) < _λ−1^λ = lim

s→0f^′(s)F (s) =α < 1 + ^N₂ for all s > 0.

• F (s)^−N2 ≲ s^N2(λ−1)+ s^N2(λ+1)e

N 2s2

for all s > 0.

Corollary 2

∫

R^N

(|u0(x )|^N2(λ−1)+|u0(x )|^N2(λ+1)e^N2|u0|2 )

dx << 1

⇒ ∃u: global sol. of (P3) s.t. lim

t→0∥u(t) − e^t∆u0∥∞= 0 and

tlim→0

[ |u(t) − u⁰|^N2(λ+1)e^{N2|u(t)−u0|2}

L¹_ul +∥u(t) − u0∥_L^N₂(λ−1)

]

= 0.

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Thank you for your kind attention!!

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