# Blowup of solutions for nonlinear nonlocal heat equations

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## Abstract

Blowup analysis for solutions of a general evolution equation with nonlocal diffusion and localized source is performed. Sufficient conditions for blowup are expressed in terms of some Morrey space norms. A comparison of these with recent results on global-in-time solutions is discussed.

## Keywords

Nonlinear nonlocal heat equation Blowup of solutions Comparison of size of global/blowing up solution## Mathematics Subject Classification

35K55 35B44## 1 Introduction

*u*with a nonnegative radially symmetric function

*J*satisfying \(\int _{\mathbb {R}^d}J(x)\,\mathrm{d}x=1\), and with the nonlinearity (a localized source) defined by a locally Lipschitz convex function \(F:[0,\infty )\rightarrow [0,\infty )\), \(F(0)=0\), satisfying the condition

Equations of the type (1) are related to the differential and integrodifferential equations (5) and (6) by their long time asymptotic behavior determined frequently by the linear equations (12)–(13) below, and studied in, e.g., [1, 11, 13].

There are plenty of results on closely related questions on conditions on the initial data guaranteeing the local-in-time existence of solutions to Eq. (5) and sufficient conditions leading to finite time blowup of solutions, see [2, 3, 14], the latter reference dealing with a general nondecreasing but not necessarily convex nonlinearity.

*mild solutions*, i.e., those satisfying the Duhamel formula

*weak solutions*so that, in particular,

*q*may exist, see [14] in the case of equations like (5), (6) with general nonlinearities. Note that this definition permits to consider locally bounded in space but unbounded solutions as in [4, 12].

Remark that unlike the case of the Eqs. (5) and (6), we cannot expect that solutions are smooth for \(t>0\), i.e., an instantaneous regularization effect of the semigroup \(\mathrm{e}^{t{{\mathcal {A}}}}\) generated by bounded diffusion operators \({\mathcal {A}}\) on solutions is absent.

*blowup*of a solution is understood here in the local \(L^\infty \) sense, i.e.,

*u*is a blowing up solution not later than at \(t=T>0\) if for some \(R>0\) the relation

Theorem 1 on local-in-time solutions to Eq. (1) that cannot be continued to global-in-time ones is one of the main results in this paper. Proposition 3 interprets a general condition in Theorem 1 in terms of the Morrey space norms related to approximative scaling properties of the problem. There are also subsidiary results on the size of global-in-time solutions compared to blowing up ones in the case of Eqs. (5) and (6).

The main idea here is that we are looking for a single quantity \(\ell =\ell (u_0)\) (a functional norm) which decides on the blowup versus global existence. Unfortunately, we do not have a dichotomic partition of the set of admissible initial data but weaker results like: \(\ell (u_0)<c\) implies the global existence while \(\ell (u_0)>C\) (with \(C>c\)) leads to a blowup of solutions in Theorem 5. Quite often the condition \(\ell (u_0)<\infty \) is necessary, but not always sufficient, for the local-in-time existence of solutions, see [4, 18] for the cases of Eqs. (5) and (6).

**Notation**The homogeneous Morrey spaces \(M^s_q(\mathbb {R}^d)\) modeled on the Lebesgue space \(L^q(\mathbb {R}^d)\), \(q\ge 1\), are defined for \(u\in L^q_{\mathrm{loc}}(\mathbb {R}^d)\) and \(1\le q\le s<\infty \), by their norms

*B*(

*x*,

*R*) denotes the ball centred at

*x*of radius

*R*: \(\{y:|y-x|<R\}\), and \(\mathbb {1}_{B(x,R)}\) is its characteristic function.

The asymptotic relation \(f\approx g\) as \(s\rightarrow s_0\) (with either \(s_0=0\) or \(s_0=\infty )\) means that \(\lim _{s\rightarrow s_0}\frac{f(s)}{g(s)}=1\), and \(f\asymp g\) is used whenever \(\lim _{s\rightarrow s_0}\frac{f(s)}{g(s)}\in (0,\infty )\). We will use the notation \(g\gg 1\) if the quantity *g* (depending on some parameters) is supposed to be sufficiently large (in terms of those parameters).

## 2 Blowup for a general nonlinear source, nonlocal diffusion model

Our aim in this paper is to give a simple proof of blowup of solutions for the Cauchy problem (1)–(2) based on the classical idea of Fujita [12]. We believe that this proof is simpler than monotonicity arguments given in [1]. Moreover, this argument applies to a class of initial data much larger than in [1, Theorem 2.4], giving explicit and rather precise general sufficient conditions on functions \(u_0\) in (2) in order to solutions of (1)–(2) blow up in a finite time, together with estimates of the blowup time, cf. also [20] for Eq. (6).

*J*are those with their Fourier transforms \({\widehat{J}}\) satisfying

*J*), and those with \(\alpha \in (0,2)\), cf. [11, Sec. 1]. Note that for \(\alpha \in (0,2)\) such

*J*do not have finite second moment. As it was studied in [11], the long time asymptotics of solutions of the linear Cauchy problem

*J*with the Fourier transform \({\widehat{J}}\) like \({\widehat{J}}(\xi )-1 \approx A|\xi |^2\log |\xi |\) as \( |\xi |\rightarrow 0\), are mentioned in [11, Th. 5.1], and then \(J(x)\asymp \frac{1}{|x|^{d+2}}\) as \(|x|\rightarrow \infty \). Thus, they are examples of operators \({\mathcal {A}}\) whose kernels have “heavy tails”: \(J(x)\asymp \frac{1}{|x|^n}\) as \(|x|\rightarrow \infty \) with some \(n>d\), discussed in [1, 11]. For them, if \(n\in (d,d+2)\), then \(\alpha =n-d\in (0,2)\) holds. Their semigroup kernels have also heavy tails unlike the Gauss-Weierstrass kernel of the heat semigroup for \(\alpha =2\). Besides bounded diffusion operators (3) studied here, the proof of Theorem 1 below applies also to unbounded operators \(\Delta \) and \(-(-\Delta )^{\alpha /2}\) as was in classical papers [12, 20].

### Theorem 1

### Proof

*h*is decreasing and satisfies \(h(0)=\infty \), \(h(\infty )=0\), by assumption (4) on the convex function

*F*. By definition (15), we have \(W_T(t)=k_{T-t}*u(.,t)(0)\) and, of course, \(z(.,t)=k_{T-t}\) solves the backward diffusion equation with the terminal condition

*J*, so that the symmetry of the semigroup, and the Jensen inequality in the last line. Integrating this from 0 to

*t*, we obtain

*every*\(R>0\) which means that

*u*blows up not later than at \(t=T\). Indeed, we have \(\lim _{t\nearrow T}\sup _{|x|>R}k_{T-t}(x)=0\) for every \(R>0\). \(\square \)

### Remark 2

We have, in this direction, the following general

### Proposition 3

### Proof

Note that for \(\mathrm{e}^{t{{\mathcal {A}}}}=\mathrm{e}^{-t(-\Delta )^{\alpha /2}}\) the assumptions on the initial data \(u_0\in L^1(\mathbb {R}^d)\) with \(\widehat{u_0}\in L^1(\mathbb {R}^d)\) can be relaxed to \(u_0\in L^1(\mathbb {R}^d)\cap L^\infty (\mathbb {R}^d)\).

Of course, condition (24) is quite general, and involves one free parameter \(T>0\). Particular examples of initial data considered in [1, Th. 2.3] leading to blowup of solutions do satisfy (24).

*R*. This kernel satisfies the bound

The proof of (ii) for \(p\le p_{\mathrm{F}}\) and \(\mathrm{e}^{t{{\mathcal {A}}}}=\mathrm{e}^{-t(-\Delta )^{\alpha /2}}\), \(\alpha \in (0,2)\), is in [20]. A rather short new proof of the result (ii) for \(\alpha =2\) and \(p=p_{\mathrm{F}}\) is in [4].

These are counterparts of results in [4, Remark 7, Theorem 2] for the classical nonlinear heat equation. These, together with results of [8, Proposition 2.3], lead to the following result, similarly as was in [4, Corollary 11] for the Cauchy problem (6) with (2). For analogous questions for radial solutions of chemotaxis systems, see also [5].

### Corollary 4

- (i)
\(|\!\!|u_0|\!\!|_{M^{d(p-1)/\alpha }_q}<c(\alpha ,d,p)\) for some \(q\in \left( 1,\frac{d(p-1)}{\alpha }\right) \), implies that problem (6) with data (2) has a global in time, smooth solution which, moreover, satisfies the time decay estimate \(\Vert u(t)\Vert _\infty ={{\mathcal {O}}}\left( t^{-1/(p-1)}\right) \).

- (ii)
\(|\!\!|u_0|\!\!|_{M^{d(p-1)/\alpha }}>C(\alpha ,d,p)\) implies that each nonnegative solution of problem (6) with the initial condition (2) blows up in a finite time.

*d*.

*u*can be continued to a global in time solution which still satisfies the bound \(0\le u(x,t)\le u_\infty (x)\).

These are natural extensions of properties of the Cauchy problem (5) with (2) studied in, e.g., [4, 17, 18].

Once again, here it should be stressed on the fact that conditions on initial data guaranteeing local-in-time existence of solutions of Eq. (5) derived in [2] but motivated by [3], and then interpreted in [4] as a bound on the Morrey space \(M^{d(p-1)/2}(\mathbb {R}^d)\) norm, are qualitatively close to those above even if they involve \(M^s_q(\mathbb {R}^d)\) spaces with any \(q>1\), see also [4, Remark 3.4], [18, Proposition 6.1].

## 3 Estimates of discrepancy

Similarly to the considerations in [9] on blowup for radial solutions of chemotaxis systems, we determine asymptotic (with respect to the variable of dimension \(d\rightarrow \infty \)) discrepancy between bounds in sufficient conditions for blowup either in terms of multiple of the singular solution or in terms of critical value of the radial concentration (and therefore of the Morrey norm) of the initial data for the model problem (6) with data (2).

### Theorem 5

(i) For each \(\alpha \in (0,2]\) and \(p>1+\frac{\alpha }{d}\) there exists a constant \(\nu _{\alpha ,p}\) independent of the dimension *d* such that if \(N>\nu _{\alpha ,p}\), then each solution of the Cauchy problem (6) with data (2) in \(\mathbb {R}^d\) with the initial data \(u_0(x)\ge N u_\infty (x)\) blows up in a finite time.

*d*such that if the \(\frac{d(p-1)}{2}\)-radial concentration of \(u_0\ge 0\) defined by

*d*such that if the \(\frac{d(p-1)}{\alpha }\)-radial concentration of \(u_0\ge 0\) defined by

### Remark 6

*d*, cf. [6, Proposition 7.1]).

### Proof

*independent*of

*d*. For \(\alpha =2\) we simply have \(c_{2,p}=\left( \frac{1}{p-1}\right) ^{\frac{1}{p-1}}\), see [4].

*independent*of

*d*. In fact, the subordinators \(f_{t,\alpha }\) satisfy \( \mathrm{e}^{-ta^\alpha }=\int _0^\infty f_{t,\alpha }(\lambda )\mathrm{e}^{-\lambda a}\,\mathrm{d}\lambda , \) so that they have selfsimilar form \(f_{t,\alpha }(\lambda )=t^{-\frac{1}{\alpha }}f_{1,\alpha }\left( \lambda t^{-\frac{1}{\alpha }}\right) \).

*d*-independent.

*independent*of

*d*. Indeed, for the normalized Lebesgue measure \(\mathrm{d}S\) on the unit sphere \({{\mathbb {S}}}^{d-1}\) we have an upper bound for the quantity

*d*. Indeed,

*different*from its counterpart for \(\alpha =2\). Now, it is clear that a sufficient condition for blowup is satisfied if

## 4 Concluding remarks

The classical idea of the proof of blowup of solutions by Fujita [12] has been applied in the wider context of problems with general linear diffusion operators and convex nonlinearities leading to qualitatively simple sufficient conditions for blowup in problems with nice approximative scaling properties.

For more classical problems involving diffusions defined by either the Laplacian or its fractional powers, these conditions have been compared with results guaranteeing the existence of global solutions. The discrepancies between bounds on quantities determining local/global behavior of solutions have been estimated.

## Notes

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