Blowup versus global in time existence of solutions for nonlinear heat equations

TL;DR

This paper presents a straightforward proof of solution blowup for nonlinear heat equations, using Morrey space norms, complementing existing conditions for global existence, and extending Fujita's method to other nonlinear parabolic equations.

Contribution

It introduces a simple blowup criterion based on Morrey space norms and extends Fujita's approach to a broader class of nonlinear parabolic equations.

Findings

Blowup occurs under specific Morrey space norm conditions.

The criterion complements existing global existence conditions.

Method extends Fujita's approach to other nonlinear equations.

Abstract

This note is devoted to a simple proof of blowup of solutions for a nonlinear heat equation. The criterion for a blowup is expressed in terms of a Morrey space norm and is in a sense complementary to conditions guaranteeing the global in time existence of solutions. The method goes back to H. Fujita and extends to other nonlinear parabolic equations.

Equations

$$u_t$$

$$u(x,0)$$

$$u_C(x)=c|x{|}^{-\frac{2}{p-1}}$$

$$c={\left(\frac{2}{p-1}\left(d-\frac{2p}{p-1}\right)\right)}^{\frac{1}{p-1}}$$

$$c^{p-1}=\gamma (d-p\gamma )=\gamma (d-2-\gamma ).$$

$$0\le u_0(x)<\delta u_C(x)=\delta c|x{|}^{-\frac{2}{p-1}}$$

$$0\le u(x,t)<\delta u_C(x)$$

$$z(x,t)=|x{|}^{\frac{2}{p-1}}u(x,t).$$

$$\nabla z(x_0,t_0)=0,\Delta z(x_0,t_0)\le 0,$$

$$\displaystyle\frac{\partial}{\partial t}z(x_{0},t)\big{|}_{t=t_{0}}$$

$$\nabla (|x{|}^{-\gamma }z)$$

$$\Delta (|x{|}^{-\gamma }z)$$

$$\underset{T>0}{\sup }{T}^{\frac{1}{p-1}}{\Vert {\mathrm{e}}^{T\Delta }{u}_0\Vert }_{\infty }>{\left(\frac{1}{p-1}\right)}^{\frac{1}{p-1}}.$$

$$G_t+\Delta G=0,G(.,T)={\delta }_0.$$

$$G(x,t)=(4\pi (T-t){)}^{-\frac{d}{2}}\exp \left(-\frac{|x{|}^2}{4(T-t)}\right),$$

$${\int }_{{\mathbb{R}}^d}G(x,t)\mathrm{d}x=1\mathrm{for}\mathrm{each}t\in [0,T).$$

$$W(t)={\int }_{{\mathbb{R}}^d}G(x,t)u(x,t)\mathrm{d}x={\mathrm{e}}^{(T-t)\Delta }u(t)(0),$$

$$W(0)\in [0,\infty )\mathrm{for}{u}_0\ge 0,\mathrm{and}W(0)\in (0,\infty )\mathrm{if}0\le u_0\not\equiv 0$$

$$W(0)={\mathrm{e}}^{T\Delta }{u}_0(0).$$

$$\frac{\mathrm{d}W(t)}{\mathrm{d}t}$$

$$\frac{\mathrm{d}W(t)}{\mathrm{d}t}\ge (W(t){)}^p$$

$$W(0{)}^{1-p}-W(t{)}^{1-p}\ge (p-1)t$$

$$W(t)\ge {\left(W(0{)}^{1-p}-(p-1)t\right)}^{-\frac{1}{p-1}}$$

$$W(0)={\mathrm{e}}^{T\Delta }{u}_0(0)>{\left((p-1)T\right)}^{-\frac{1}{p-1}},$$

$$T^{\frac{1}{p-1}}{\mathrm{e}}^{T\Delta }{u}_0(0)>{\left(\frac{1}{p-1}\right)}^{\frac{1}{p-1}}$$

$$\underset{T>0}{\sup }{T}^{\frac{1}{p-1}}\Vert {\mathrm{e}}^{T\Delta }{u}_0{\Vert }_{\infty }$$

$$(4\pi {)}^{-\frac{d}{2}}{\int }_{{\mathbb{R}}^d}{\mathrm{e}}^{-|y{|}^2/4T}{u}_0(y)\mathrm{d}y>{\left(\frac{1}{p-1}\right)}^{\frac{1}{p-1}},$$

$$u(t)={\mathrm{e}}^{t\Delta }{u}_0+{\int }_0^t{\mathrm{e}}^{(t-\tau )\Delta }u(\tau {)}^p\mathrm{d}\tau .$$

$$u(x,T)-{\mathrm{e}}^{T\Delta }{u}_0(x)$$

$$\Vert u(T){\Vert }_1$$

$$\Vert u(t+T){\Vert }_1\ge \Vert u(t){\Vert }_1\left(1+\widetilde{\stackrel{~}{c}}\Vert u_0{\Vert }^{p-1}\right)$$

Full text

The author, partially supported by the NCN grant 2013/09/B/ST1/04412, thanks Ignacio Guerra for interesting conversations leading to revisiting Fujita’s method and Philippe Souplet for many pertinent remarks.

In memory of Marek Burnat

submitted to the special volume of Topological Methods in Nonlinear Analysis

Introduction

We consider in this paper the Cauchy problem for the simplest example of semilinear parabolic equation in $\mathbb{R}^{d}$, $d\geq 1$, $p>1$,

[TABLE]

This problem has been thoroughly studied beginning with [9, 11, 12], and many fine properties of its solutions are known. For the reference, see the extensive monograph [21] and, in particular, a recent paper [22].

The purpose of this note is to give a short proof of nonexistence of global in time (and sometimes also local in time) positive solutions to problem (1)–(2) together with criteria for blowup expressed in terms of the norms of critical Morrey spaces. Those criteria are, in a sense, complementary to assumptions guaranteeing the global in time existence of solutions. The idea of the proof goes back to the seminal paper [9] but the straightforward connection to the Morrey spaces norms seems to be missed out up to now. An idea in [8, proof of Th. 3] in the context of radial solutions of the Keller-Segel model of chemotaxis is also reminiscent of that. Recently, this approach has been systematically developed for (radially symmetric solutions of) the chemotaxis models with different diffusion operators in [7] as an alternative to other proofs based on considerations of (new) moments in [3, 4, 5, 2, 6]. Here, for the nonlinear heat equation generally we do not use geometric assumptions on solutions such as radial symmetry. This method classically introduced by [9] is flexible enough, and extends also to other nonlinear parabolic problems, see remarks at the end of this paper.

Whenever problem (1)–(2) admits a singular stationary solution, this plays an important role in determining when solutions with $u_{0}$ featuring singularities either lead to global in time solutions or they blow up in a finite time. The form of this solution is well known, we recall this below.

[TABLE]

with the constant

[TABLE]

is a stationary positive (weak and pointwise) solution of equation (1).

[TABLE]

Since $p\gamma<d$, this is a distributional solution of equation (1). $\square$

A typical example of a global in time existence result is the following

[TABLE]

for some $\delta<1$. Then any solution $u\in{\mathcal{C}}^{2}(\mathbb{R}^{d}\times[0,T])$ of problem (1)–(2) with the property $\lim_{|x|\to\infty}|x|^{\frac{2}{p-1}}u(x,t)=0$ satisfied uniformly on $[0,T]$, exists globally in time and is bounded by $u_{C}$

[TABLE]

for each $t>0$ and $x\in\mathbb{R}^{d}$.

We sketch the proof skipping some technical details related to the local existence of solutions with the assumed regularity and decay. Suppose a contrario that $u(x_{0},t_{0})=\delta u_{C}(x_{0})$ for some $x_{0}\in\mathbb{R}^{d}$ with minimal $|x_{0}|$ and the least $t_{0}>0$. Consider the auxiliary function

[TABLE]

Under the a contrario assumption $z(x_{0},t_{0})=\delta c$,   and if $t_{0}$ is the first moment when $z(x,t)$ hits the level $\delta c\in\mathbb{R}$, we have

[TABLE]

since $x_{0}$ is the point where the maximum of $z$ (equal to $\delta c$) is attained. Let us compute

[TABLE]

since

[TABLE]

relations (9) hold, and recall (5) to see that $\gamma(\gamma+2-d)=-c^{p-1}$. The inequality \frac{\partial}{\partial t}z(x_{0},t)\big{|}_{t=t_{0}}<0 contradicts the assumption that $z$ hits for the first time the constant level $\delta c$ at $t=t_{0}$. $\square$

Solutions exploding in a finite time

Our goal here is to show a finite time blowup, and that the critical size of some functional norm of initial data leading to a blowup is close to the optimal size of this norm guaranteeing the existence of a global in time regular positive solution. The first argument is essentially that of [9]. The considerations in [11, 12] employed some (quite complicated) moments and energies of solutions with Gaussian weights but not exactly quantity (15).

[TABLE]

Then, any local in time weak solution $u=u(x,t)$ of the Cauchy problem (1)–(2) with $p>1$ cannot be continued beyond $t=T$.

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Clearly, we have a solution

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the unique nonnegative one, satisfying moreover

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We consider $t\in[0,T)$, and define for a solution $u$ of (1)–(2) which is supposed to exist on $[0,T)$ the moment

[TABLE]

where ${\rm e}^{t\Delta}$, $t>0$, denotes the heat semigroup on $\mathbb{R}^{d}$ defined with the Gauss–Weierstrass kernel. Evidently, we have

[TABLE]

and, moreover,

[TABLE]

Since $G$ decays exponentially in $x$ as $|x|\to\infty$, the moment $W$ is well defined (at least) for (weak, pointwise, distributional) solutions $u=u(x,t)$ which are polynomially bounded in $x$ as $|x|\to\infty$. The evolution of the moment $W$ is governed by the identity

[TABLE]

where the third line means that $u$ is a weak solution, and the last line follows by the Hölder inequality and property (14). Now, the differential inequality

[TABLE]

(with the strict inequality for nonconstant functions $u(t)$) shows that $W(t)$ increases, and after integrating immediately leads to

[TABLE]

for all $t\in[0,T)$, so that

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again for all $t\in[0,T)$. Now, passing to the limit $t\nearrow T$, it is clear that if

[TABLE]

then we arrive at a contradiction with property (16), so that the solution blows up not later than $t=T$. In other words, if the condition

[TABLE]

is satisfied, then the solution $u$ cannot exist for $t=T$. By the translational invariance of equation (1), for positive $u_{0}$ condition (22) is equivalent to inequality (11). $\square$

[TABLE]

so a sufficient condition (11) for blowup holds. Here, we offer a proof of the analogous result if $d=\frac{2}{p-1}$, an alternative to the one given in [26, Th. 1]. Since condition (11) for $\frac{d}{2}=\frac{1}{p-1}$ reads

[TABLE]

and $\lim_{T\to\infty}W(0)=(4\pi)^{-\frac{d}{2}}\|u_{0}\|_{1}$, it suffices to show that $\|u(t)\|_{1}$ becomes large enough for some $t\geq 0$, and replace the initial condition $u_{0}$ by $u(t)$.

Problem (1)–(2) for positive solutions can be rewritten in the mild form as

[TABLE]

Let us write the Duhamel formula (24) for $t=T$ in a more detailed way as

[TABLE]

by the Hölder inequality and by the Cauchy inequality $\frac{1}{2}\left({\rm e}^{x\cdot y/2(T-\tau)}+{\rm e}^{-x\cdot y/2(T-\tau)}\right)\geq 1$. Now, integrating over $x\in\mathbb{R}^{d}$, we obtain as a consequence of (6)

[TABLE]

for some $\tilde{\tilde{c}}$ independent of $u_{0}$. Observe that the norm $\|u(t)\|_{1}$ increases in time. Therefore, by a shift of time, we have

[TABLE]

and it is clear that for some $t$ the norm $\|u(t)\|_{1}$ becomes large enough in order to condition (11) holds with time shifted from [math] to $t$, see also (23). Therefore, $u(t)$ blows up in a finite time. $\square$

[TABLE]

Caution: the notation for Morrey spaces used elsewhere might be different, e.g. $M^{s}_{q}$ is denoted by $M^{q,\lambda}$ with $\lambda=dq/s$ in [22].

The most frequent situation is when $q=1$ and we consider $M^{s}_{1}\equiv M^{s}$. The spaces $M^{d(p-1)/2}(\mathbb{R}^{d})$ and $M^{d(p-1)/2}_{q}(\mathbb{R}^{d})$, $q>1$, are critical (i.e. invariant by scalings that conserve equation (1)) in the study of equation (1), see [22], and we refer to [1, 5, 7, 14] for analogous examples in chemotaxis theory.

For instance, if the norm $|\!\!|u_{0}|\!\!|_{M^{d(p-1)/2}_{q}}$ (for a number $q>1$) is small enough, then a solution of problem (1)–(2) is global in time, see [22, Proposition 6.1] (and for the chemotaxis system cf. [1, Th. 1]). The former result can be proved directly (while the proof in [22] was by contradiction) using the Picard iterations of the mapping

[TABLE]

$u_{n+1}={\mathcal{N}}(u_{n})$, $n=1,\,2,\,\dots$, with $u_{0}\in M^{d(p-1)/2}_{q}(\mathbb{R}^{d})$ ($q>1$, $p>1+2q/d$) small enough. They are convergent in the norm $|\!|\!|u(t)|\!|\!|=\sup_{t>0}t^{(1-q/r)/(p-1)}|\!\!|u(t)|\!\!|_{M^{rd(p-1)/(2q)}_{r}}$ for $\max\{p,q\}<r<pq$, since $|\!|\!|{\rm e}^{t\Delta}u_{0}|\!|\!|<\infty$.

Taking into account the above remarks on the critical Morrey space, we formulate the following partial dichotomy result

• (i)

$|\!\!|u_{0}|\!\!|_{M^{d(p-1)/2}_{q}}<c_{d}$* implies that problem (1)–(2) has a global in time solution;*

• (ii)

$|\!\!|u_{0}|\!\!|_{M^{d(p-1)/2}}>C_{d}$* implies that each solution of problem (1)–(2) blows up in a finite time.*

Below, we determine two threshold numbers measuring how big must be either $u_{0}$ compared to $u_{C}$ or the Morrey norm $M^{d(p-1)/2}(\mathbb{R}^{d})$ of a radial $u_{0}$ in order to a solution of (1)–(2) blows up in a finite time.

[TABLE]

Then, the relation

[TABLE]

holds. More precisely,

[TABLE]

as $d\to\infty$.

(ii) Moreover, there exists $\mathcal{M}>0$ such that if $u_{0}\geq 0$ is radially symmetric and such that $|\!\!|u_{0}|\!\!|_{M^{d(p-1)/2}}>{\mathcal{M}}$, then the solution with $u_{0}$ as the initial data blows up in a finite time. A rough estimate from above for the threshold value of ${\mathcal{M}}$ when $p>\frac{d+2}{d}$ (i.e. exactly when $d-\gamma>0$) is the following

[TABLE]

as $d\to\infty$.

[TABLE]

where we used the fact that the area $\sigma_{d}$ of the unit sphere in $\mathbb{R}^{d}$ is equal to

[TABLE]

and we obtain (27). The asymptotic formula in (28) is a consequence of the Stirling formula for the Gamma function

[TABLE]

see e.g. [25]. Indeed,

[TABLE]

holds, i.e. the discrepancy between the multiples of $|x|^{-\frac{2}{p-1}}$ in the initial data corresponding to global and blowing up solutions converges to $1$ as $d\to\infty$.

(ii) Define for a radially symmetric function $u_{0}\geq 0$ its radial distribution function

[TABLE]

so that $\sigma_{d}u_{0}(r)r^{d-1}=M_{r}(r)$. It is not hard to see that for radial $u_{0}\geq 0$ we have $|\!\!|u_{0}|\!\!|_{M^{d(p-1)/2}}=\sup_{R>0}R^{\gamma-d}\int_{\{|x|<R\}}u_{0}(x)\,{\rm d}x$. If $u_{0}\in M^{d(p-1)/2}(\mathbb{R}^{d})$ with $|\!\!|u_{0}|\!\!|_{M^{d(p-1)/2}}={\mathcal{M}}$, then for each $\varepsilon>0$ there exists $R>0$ such that $M(R)\geq({\mathcal{M}}-\varepsilon)R^{d-\gamma}$. We need to estimate $T^{\frac{1}{p-1}}W(0)=T^{\frac{1}{p-1}}{\rm e}^{T\Delta}u_{0}(0)$ from below, and we do it in a rough way as

[TABLE]

Since $\max_{T>0}T^{\frac{1}{p-1}-\frac{d}{2}}{\rm e}^{-R^{2}/4T}$ is attained when $\frac{R^{2}}{4T}=\frac{1}{2}(d-\gamma)$ and this quantity is equal to $\left(\frac{R^{2}{\rm e}}{2(d-\gamma)}\right)^{\frac{\gamma-d}{2}}$, we need to determine when

[TABLE]

This leads, by (32) and (33), to

[TABLE]

Note that, compared to $|\!\!|u_{C}|\!\!|_{M^{d(p-1)/2}}=\frac{c\sigma_{d}}{d-\gamma}$, the number ${\mathcal{M}}$ is much bigger:

[TABLE]

and, of course, ${\mathcal{M}}\leq C_{d}$ by the definition of $C_{d}$ in Corollary 10 (ii). $\square$

A very short proof of the former result follows from the observation that such an initial condition $u_{0}$ satisfies $|\!\!|u_{0}|\!\!|_{M^{d(p-1)/2}}=\infty$, hence condition (11) holds. Indeed, inequality

[TABLE]

is satisfied with arbitrarily large constants $C$ and suitably big fixed $R_{0}$, so that

$\lim_{R\to\infty}R^{\gamma-d}\int_{|x|\leq R}u_{0}(x)\,{\rm d}x\geq C$.

Extensions and generalizations. The method of proving blowup with the use of the Gaussian moment (15) extends to other nonlinear equations with the heat operator in the principal part. For instance, the equation

[TABLE]

considered in [15] leads to the following sufficient for blowup of its positive solutions

[TABLE]

These integrals can be calculated explicitly, e.g. for $Q(x)=|x|^{\beta}$, $\beta\in\mathbb{R}$.

Another example for which sufficient criteria for blowup can be easily established with this method is the quasilinear equation

[TABLE]

and, of course, radially symmetric solutions of the Keller-Segel system in chemotaxis theory as it was mentioned before.