Global existence and blowup for a parabolic equation with a non-local source and absorption
The authors consider the following one-dimensional free boundary problem for a parabolic equation with a non-local source and absorption: \[ u_{t}-du_{xx}=\int_{g(t)}^{h(t)}u^{p}(t,x)dx-ku^{q}(t,x),\quad t>0,\;g(t)<x<h(t), \] \[ u(t,g(t))=0,\quad g'(t)=-\mu u_{x}(t,g(t)),\quad t>0 \] \[ u(t,h(t))=0,\quad h'(t)=-\mu u_{x}(t,h(t)),\quad t>0 \] \[ g(0)=-h_{0},\quad h(0)=h_{0},\quad u(0,x)=u_{0}(x),\quad -h_{0}\leq x\leq h_{0}, \] where both \(x=g(t)\) and \(x=h(t)\) are moving boundaries to be determined together with the solution \(u(t,x).\) Here \(h_{0}>0,\) \(p>q>1,\) \(d,k\) and \(\mu\) are positive constants, and the initial function \(u_{0}\) is a non-negative non-trivial function satisfying \(u_{0}\in C^{2}([-h_{0}h_{0}]),\) \(u_{0}(-h_{0})=u_{0}(h_{0})=0\) and \(u_{0}>0\) in \((-h_{0},h_{0}.)\) The local existence and uniqueness is investigated. The description of the free boundaries is given. As main result of the blowup property discussion consists in the proof of the existence of a global fast solution and a global slow solution.
scientific article; zbMATH DE number 6153126
scientific article