A New Necessary and Sufficient Condition for the Existence of Global Solutions to Semilinear Parabolic Equations on Bounded Domains

DOI10.1016/J.CHAOS.2022.112055arXiv2207.08383MaRDI QIDQ6405275

Soon-Yeong Chung, Jaeho Hwang

Publication date: 18 July 2022

Abstract: The purpose of this paper is to give a necessary and sufficient condition for the existence and non-existence of global solutions of the following semilinear parabolic equations [ u_{t}=Delta u+psi(t)f(u),,,mbox{ in }Omega imes (0,t^{*}), ] under the Dirichlet boundary condition on a bounded domain. In fact, this has remained as an open problem for a few decades, even for the case $f(u)=u^p$. As a matter of fact, we prove: [ �egin{aligned} &mbox{there is no global solution for any initial data if and only if } &mbox{the function } f mbox{ satisfies} &hspace{20mm}int_{0}^{infty}psi(t)frac{fleft(lVert S(t)u_{0} Vert_{infty} ight)}{lVert S(t)u_{0} Vert_{infty}}dt=infty &mbox{for every },epsilon>0,mbox{ and nonnegative nontrivial initial data },u_{0}in C_{0}(Omega). end{aligned} ] Here, $(S(t){)}_{tgeq0}$ is the heat semigroup with the Dirichlet boundary condition.

Mathematics Subject Classification ID

Semilinear parabolic equations (35K58) PDEs on graphs and networks (ramified or polygonal spaces) (35R02)

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