Solvability conditions for some semi-linear parabolic equations
The authors deal with the Cauchy problem for a semi-linear parabolic equation of the following form: \[ \begin{cases} \Delta u-\partial _tu+Vu^p+w =0,\;p>1 \\ u(x,0) =u_0(x)\end{cases}\tag{1} \] where \(u_0(x)\geq 0\), \(w=w(x)\), \(V=V(x)\), \(u_0,w,V\in L_{\text{loc}}^1(\mathbb R^N)\), \(x\in \mathbb R^N\), \(t>0\). Starting from the fact that the solvability of problem (1) was for a long time a open problem, the authors establish necessary conditions on the potential \(V\) so that problem (1) has a positive solution for some \(u_0=u_0(x)\geq 0\), \(w=w(x)\geq 0\). Finally the authors give an application in the case when \(V\) is the important inverse square potential.
scientific article; zbMATH DE number 5625448
scientific article