9,130 research outputs found
Lower bounds for the blow-up time in a non-local reaction–diffusion problem
AbstractFor a non-local reaction–diffusion problem with either homogeneous Dirichlet or homogeneous Neumann boundary conditions, the questions of blow-up are investigated. Specifically, if the solutions blow up, lower bounds for the time of blow-up are derived
On the lifespan of classical solutions to a non-local porous medium problem with nonlinear boundary conditions
In this paper we analyze the porous medium equation
\begin{equation}\label{ProblemAbstract} \tag{} %\begin{cases}
u_t=\Delta u^m + a\io u^p-b u^q -c\lvert\nabla\sqrt{u}\rvert^2 \quad
\textrm{in}\quad \Omega \times I,%\\ %u_\nu-g(u)=0 & \textrm{on}\; \partial
\Omega, t>0,\\ %u({\bf x},0)=u_0({\bf x})&{\bf x} \in \Omega,\\ %\end{cases}
\end{equation} where is a bounded and smooth domain of , with
, and is the maximal interval of existence for . The
constants are positive, proper real numbers larger than 1 and
the equation is complemented with nonlinear boundary conditions involving the
outward normal derivative of . Under some hypothesis on the data, including
intrinsic relations between and , and assuming that for some positive
and sufficiently regular function u_0(\nx) the Initial Boundary Value Problem
(IBVP) associated to \eqref{ProblemAbstract} possesses a positive classical
solution u=u(\nx,t) on : \begin{itemize} \item
[] when and in 2- and 3-dimensional domains, we determine
a \textit{lower bound of} for those becoming unbounded in
at such ; \item [] when and in
-dimensional settings, we establish a \textit{global existence criterion}
for . \end{itemize
Nonlinear Convection in Reaction-diffusion Equations under dynamical boundary conditions
We investigate blow-up phenomena for positive solutions of nonlinear
reaction-diffusion equations including a nonlinear convection term in a bounded domain of
under the dissipative dynamical boundary conditions . Some conditions on and are discussed
to state if the positive solutions blow up in finite time or not. Moreover, for
certain classes of nonlinearities, an upper-bound for the blow-up time can be
derived and the blow-up rate can be determinated.Comment: 20 page
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