Capacitary estimates of solutions of semilinear parabolic equations
This paper is concerned with estimates for solutions of the semilinear parabolic equation \(u_t-\Delta u+u^q=0\) in \(\mathbb R^N\times (0,\infty)\), \(q>1\). The authors obtain that any positive solution for this equation with initial trace \((F,0)\), where \(F\subset\mathbb R^N\) is a closed set, can be represented up to two universal multiplicative constants by a series involving the \(C_{2/q,q'}\)-Bessel capacity. As a consequence, the uniqueness of such a solution is deduced. Next, the authors characterize the blow-up set of \(u(x,t)\) as \(t\to 0\) using the density of the set \(F\) expressed in terms of the \(C_{2/q,q'}\)-Bessel capacity.
scientific article; zbMATH DE number 6210451