Capacitary estimates of solutions of semilinear parabolic equations

2010/09/24 by Marcus, Moshe, Veron, Laurent
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1009.4848

Abstract

We prove that any positive solution of \prt_tu-Δu+u^q=0 (q>1) in \BBR^N\ti(0,∞) with initial trace (F,0), where F is a closed subset of \BBR^N can be estimated from above and below and up to two universal multiplicative constants, by a series involving the Bessel capacity C_2/q,q'. As a consequence we prove that there exists a unique positive solution of the equation with such an initial trace. We also characterize the blow-up set of u(x,t) when t\downarrow 0, by using the "density" of F expressed in terms of the C_2/q,q'-capacity.

Related

• Capacitary estimates of solutions of semilinear parabolic equations
• Initial traces and solvability of Cauchy problem to a semilinear parabolic system
• Pointwise Bounds and Blow-up for Systems of Semilinear Parabolic Inequalities and Nonlinear Heat Potential Estimates
• Threshold and strong threshold solutions of a semilinear parabolic equation
• Initial Blow-up of Solutions of Semilinear Parabolic Inequalities