Proceedings of the Royal Society of Edinburgh Section A Mathematics · 1997 · 253 citations · 8 references
Natural SciencesMathematicsDifferential EquationsStochastic ProcessesTopologyBoundary ConditionsBoundary-value ProblemsExistence TheoryInitial Value ProblemsOperator TheoryPartial Differential EquationsMonotone Operator Existence TheoremsUniqueness ConditionsUniqueness In Nonlinear Differential EquationsUniqueness In Parabolic EquationsUniqueness Of Solutions
In this paper we prove the existence and uniqueness of a renormalised solution of the nonlinear problem where the data f and u 0 belong to L 1 (Ω × (0, T )) and L 1 (Ω), and where the function a :(0, T ) × Ω × ℝ N → ℝ N is monotone (but not necessarily strictly monotone) and defines a bounded coercive continuous operator from the space into its dual space. The renormalised solution is an element of C 0 ([ 0, T ] L 1 (Ω)) such that its truncates T K (u) belong to with this solution satisfies the equation formally obtained by using in the equation the test function S(u)φ , where φ belongs to and where S belongs to C ∞ (ℝ) with
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