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Blow-up for a heat equation with convection and boundary flux

Published online by Cambridge University Press:  14 July 2008

Arturo de Pablo
Affiliation:
Departamento de Matemáticas, \onm{Universidad Carlos III de Madrid}, 28911 Leganés, Spain (arturop@math.uc3m.es)
Guillermo Reyes
Affiliation:
Departamento de Matemáticas, Universidad Politécnica de Madrid, 28040 Madrid, Spain (greyes@caminos.upm.es)
Ariel Sánchez
Affiliation:
Departamento de Matemáticas, Universidad Rey Juan Carlos, 28933 Móstoles, Spain (ariel.sanchez@urjc.es)

Abstract

We study the evolution of solutions to the initial-boundary-value problem

\begin{alignat*}{3} u_t&=(u^m)_{xx}+\lambda(u^q)_x, & \quad x&>0,&\quad t&\in(0,T), \\[2pt] -(u^m)_x(0,t)&=u^p(0,t), & &&t&\in(0,T), \\[2pt] u(x,0)&=u_0(x), & \quad x&>0, \end{alignat*}

and give a rather complete characterization, in terms of the parameters $m\ge1$, $p,q>0$ and $\lambda>0$, of whether all solutions are global in time or, on the contrary, there exist blow-up solutions. We show that the presence of the convective term has a preventive effect on the blow-up (with respect to the case $\lambda=0$) and gives rise to a collapse of the region where all solutions blow up in this case. On the other hand, a new Fujita-type phenomenon takes place at the level $p=q$ and $0<\lambda<1$.

Type
Research Article
Copyright
2008 Royal Society of Edinburgh

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