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Initial trace of positive solutions to semilinear parabolic inequalities. (English) Zbl 1021.35051

The goal of this paper is to introduce a new method for defining the initial trace of a solution to \[ \partial_t u-\Delta u+g(x,t,u)=0\text{ in }\Omega\times [0,T], \] based upon resolution of initial value problems with data measures and the notion of subcriticality. Here \(\Omega\) is a domain in \(\mathbb{R}^N\) and \(g\) is a continuous function defined in \(\Omega\times(0,T]\times \mathbb{R}\). Under some natural assumptions on \(g\), the authors prove that initial trace is a Borel measure (which may blow up on compact sets).

MSC:

35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
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