A singular semilinear parabolic equation (Q2761884)

The authors investigate the existence of distribution solutions for two problems: the Cauchy problem to semilinear heat equations with singular potential as well as singular coefficients NEWLINE\[NEWLINE u_t-t^{-\sigma }\triangle u=V(x)|u|^{p-1}u+f(x), \quad (x,t)\in \mathbb{R}^n\times (0,\infty) NEWLINE\]NEWLINE under initial condition \(u(x,t)=u_0(x)\), \(x\in \mathbb{R}^n\) and the initial-boundary value problem for the same equation in \(\Omega \times (0,T]\) with initial data in \(\Omega \) and boundary condition \(u(x,t)|_{\partial\Omega\times (0,T)}=0\), \(n\geq 3\), \(p>1\), \(\Omega \subset \mathbb{R}^n\) is a smooth domain containing the origin. With the aid of the Kato class, Green tight functions, and the \(3G\) theorem it is shown that there exist weak solutions to both problems under considerations.