Critical exponent and critical blow-up for quasilinear parabolic equations
This paper deals with nonlinear parabolic equations of the following type \[ u_t-\Delta u^m= u^p\quad\text{in }D\times (0,T), \] \[ u(x,t)= 0\quad\text{on }\partial D\times (0,T),\quad u(x,0)= u_0(x)\geq 0. \] Here \(D\) is either \(\mathbb{R}^N\) or an exterior domain. It is shown that for certain ranges of \(p\) and \(m\) no nontrivial solution exists for all times \(t>0\). If a solution ceases to exist, it blows up. The authors use a variant of the method of Fourier coefficients which is common in this type of studies. They obtain interesting estimates for the solutions.
scientific article; zbMATH DE number 1033782
scientific article