Stability of the blow-up time and the blow-up set under perturbations
José M. Arrieta Raúl Ferreira Arturo de Pablo Julio D. Rossi
Discrete & Continuous Dynamical Systems - A 2010, 26(1): 43-61 doi: 10.3934/dcds.2010.26.43
In this paper we prove a general result concerning continuity of the blow-up time and the blow-up set for an evolution problem under perturbations. This result is based on some convergence of the perturbations for times smaller than the blow-up time of the unperturbed problem together with uniform bounds on the blow-up rates of the perturbed problems.
   We also present several examples. Among them we consider changing the spacial domain in which the heat equation with a power source takes place. We consider rather general perturbations of the domain and show the continuity of the blow-up time. Moreover, we deal with perturbations on the initial condition and on parameters in the equation. Finally, we also present some continuity results for the blow-up set.
keywords: blow-up Stability perturbations.
Nonlocal heat equations: Regularizing effect, decay estimates and Nash inequalities
Cristina Brändle Arturo De Pablo
Communications on Pure & Applied Analysis 2018, 17(3): 1161-1178 doi: 10.3934/cpaa.2018056

We study the short and large time behaviour of solutions of nonlocal heat equations of the form $\partial_tu+\mathcal{L} u = 0$. Here $\mathcal{L}$ is an integral operator given by a symmetric nonnegative kernel of Lévy type, that includes bounded and unbounded transition probability densities. We characterize when a regularizing effect occurs for small times and obtain $L^q$-$L^p$ decay estimates, $1≤ q < p < ∞$ when the time is large. These properties turn out to depend only on the behaviour of the kernel at the origin or at infinity, respectively, without need of any information at the other end. An equivalence between the decay and a restricted Nash inequality is shown. Finally we deal with the decay of nonlinear nonlocal equations of porous medium type $\partial_tu+\mathcal{L}Φ(u) = 0$.

keywords: Nonlocal diffusion equations integral operators asymptotic behaviour Nash inequalities
The Cauchy problem for a nonhomogeneous heat equation with reaction
Arturo de Pablo Guillermo Reyes Ariel Sánchez
Discrete & Continuous Dynamical Systems - A 2013, 33(2): 643-662 doi: 10.3934/dcds.2013.33.643
We study the behaviour of the solutions to the Cauchy problem $$ \left\{\begin{array}{ll} \rho(x)u_t=\Delta u+u^p,&\quad x\in\mathbb{R}^N ,\;t\in(0,T),\\ u(x,\, 0)=u_0(x),&\quad x\in\mathbb{R}^N , \end{array}\right. $$ with $p>0$ and a positive density $\rho$ satisfying $\rho(x)\sim|x|^{-\sigma}$ for large $|x|$, $0<\sigma<2< N$. We consider both the cases of a bounded density and the singular density $\rho(x)=|x|^{-\sigma}$. We are interested in describing sharp decay conditions on the data at infinity that guarantee local/global existence of solutions, which are unique in classes of functions with the same decay. We prove that larger data give rise to instantaneous complete blow-up. We also deal with the occurrence of finite-time blow-up. We prove that the global existence exponent is $p_0=1$, while the Fujita exponent depends on $\sigma$, namely $p_c=1+\frac2{N-\sigma}$.
    We show that instantaneous blow-up at space infinity takes place when $p\le1$.
    We also briefly discuss the case $2<\sigma< N$: we prove that the Fujita exponent in this case does not depend on $\sigma$, $\tilde{p}_c=1+\frac2{N-2}$, and for initial values not too small at infinity a phenomenon of instantaneous complete blow-up occurs in the range $1< p < \tilde{p}_c$
keywords: blow-up Reaction-diffusion equations well-posedness initial value problem critical exponents.

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