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Proceedings of the American Mathematical Society

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On the existence of maximal and minimal solutions for parabolic partial differential equations


Authors: J. W. Bebernes and K. Schmitt
Journal: Proc. Amer. Math. Soc. 73 (1979), 211-218
MSC: Primary 35K55
DOI: https://doi.org/10.1090/S0002-9939-1979-0516467-3
MathSciNet review: 516467
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Abstract: The existence of maximal and minimal solutions for initial-boundary value problems and the Cauchy initial value problem associated with $ Lu = f(x,t,u,\nabla u)$ where L is a second order uniformly parabolic differential operator is obtained by constructing maximal and minimal solutions from all possible lower and all possible upper solutions, respectively. This approach allows f to be highly nonlinear, i.e., f locally Hölder continuous with almost quadratic growth in $ \vert\nabla u\vert$.


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DOI: https://doi.org/10.1090/S0002-9939-1979-0516467-3
Keywords: Maximal solutions, parabolic partial differential equations, lower solutions, nonuniqueness
Article copyright: © Copyright 1979 American Mathematical Society