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DCDS

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.

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.

DCDS

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 $1< p < \tilde{p}_c$

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 $1< p < \tilde{p}_c$

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