Abstract
We prove existence of solutions for parabolic initial value problems @tu = u + f(u) on R N , where f : R! R is a bounded, but possibly discontinuous function. AMS Classication: 35K57
Highlights
We prove an existence theorem for the following parabolic initial value problem
Similar problems were investigated by several authors mainly on bounded domains
The equation is usually considered as a differential inclusion
Summary
One of the first results in this field was achieved by Rauch [14] He proved the existence of a solution u ∈ L2([0, t∗], H01(Ω)), where Ω ⊂ RN is a bounded domain and f is locally bounded. We prove that there exists a continuous solution on RN × [0, ∞). Definition 1 The function u : Q = RN × [0, ∞) → R is called a solution of (1)–(2) if (i) u ∈ C1,0(Q), that is u is continuous in Q and continuously differentiable w.r.t. x in Q (ii) u satisfies the corresponding differential inclusion in the weak sense, that is there exists a bounded measurable function h : Q → R such that (u∂tφ − ∇u, ∇φ + hφ) = 0 for all φ ∈ C0∞(Q).
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