Blow-up of nonnegative solutions to quasilinear parabolic inequalities (Q5926047)

The authors investigate blow-up of nonnegative solutions to parabolic inequalities of the following type: NEWLINE\[NEWLINE\rho(x,t) \partial_t (u^k)\geq \sum^n_{i,j=1} \partial_{x_i} \biggl[a_{ij} (x,t,u)f \bigl(|\nabla u|\bigr) \partial_{x_j} u\biggr]+ c(x,t,u)u^qNEWLINE\]NEWLINE in \(\mathbb{R}^n\times (0,+\infty)\); here \(k>0\), \(q>1\) and \(\rho,a_{ij},f,c\) are given functions. The purpose is to investigate critical exponents for nonexistence of global solutions to the evolution problem. The authors derive sufficient conditions, which ensure nonexistence of global nonnegative solutions and, in particular, they determine the critical exponent for the corresponding Cauchy problem.NEWLINENEWLINENEWLINEThe approach leads to the concept of nonlinear capacity.