Existence and comparison theorems for nonlinear diffusion systems

Hendrik J. Kuiper

Research output: Contribution to journalArticle

10 Scopus citations

Abstract

In this paper we prove existence, uniqueness, and regularity results for systems of nonlinear second order parabolic equations with boundary conditions of the Dirichlet, Neumann, and regular oblique derivative types. Let K(t) consist of all functions (v1(x), v2(x),..., vm(x)) from Ω ⊂ Rn into Rm which satisfy ψi(x, t) ≤ vi(x) ≤ θi(x, t) for all x ε{lunate} Ω and 1 ≤ i ≤ m, where ψiand θi are extended real-valued functions on \ ̄gW × [0, T). We find conditions which will ensure that a solution U(x, t) ≡ (u1(x, t), u2(x, t),..., um(x, t)) which satisfies U(x, 0) ε{lunate} K(0) will also satisfy U(x, t) ε{lunate} K(t) for all 0 ≤ t < T. This result, which has some similarity to the Gronwall Inequality, is then used to prove a global existence theorem.

Original languageEnglish (US)
Pages (from-to)166-181
Number of pages16
JournalJournal of Mathematical Analysis and Applications
Volume60
Issue number1
DOIs
StatePublished - Aug 1977

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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