Asymptotic analysis for a nonlinear parabolic equation on \(\mathbb R\) (Q2713590)

Let \(u(t,x)\) be a nonnegative solution of the equation \(u_t-u_{xx}+f(u)=0\) satisfying the initial data \(au_0\), where \(a\) is a nonnegative parameter and the nonnegative function \(u_0(x)\) has a compact support. Under some assumptions on the sublinear function \(f\), the author proves the existence of a critical number \(a_c\) such that the solutions either converge to zero (for \(a<a_c\)), blow up in \(L^2\)-norm (for \(a>a_c\)) or converge to an even positive stationary solution (for \(a=a_c\)), as \(t\to \infty \). The second part of the article is devoted to some results of the problem in the case of a superlinear function \(f(u).\)