Critical exponents for the decay rate of solutions in a semilinear parabolic equation
The authors study the 1-D Cauchy problem for the heat equation with power-type absorption \[ u_t=u_{xx}-|u|^{p-1}u \] with \(u(x,0)= u_0(x)\) uniformly bounded. Then \(\max_{x\in \mathbb{R}}|u(x,t) |t^{1/(p-1)}\) is uniformly bounded, and a solution is called to decay fast, if this quantity tends to zero. The authors show, that this cannot happen, if \(u_0\) changes sign exactly \(k\)-times and \(1<p<p_k =1+2/(k+1)\), while for \(p\geq p_k\) such an initial value leading to a fast decaying solution will exist. Earlier, the authors had proved, that the same exponents occur in the case of finite time blow-up if dealing with a source term ``\(+|u|^{p-1}u\)''.
scientific article; zbMATH DE number 1287909