A Nonexistence Theorem for the Heat Equation with a Nonlinear Boundary Condition and for the Porous Medium Equation Backward in Time.
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WISCONSIN UNIV MADISON MATHEMATICS RESEARCH CENTER
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The paper deals with two nonlinear problems for parabolic equations. The first, problem A, is an initial-boundary value problem for the heat equation where the nonlinearity is in the boundary condition. The second, problem B, is a final value problem for the porous medium equation. It is shown that if the nonlinearity and initial data in A satisfy certain restrictions then no classical or weak solution of A can exist for all time. It is further shown that no weak solution of B can have existed for all previous time. An indication is given of how the methods used in A can be used to obtain under reasonable hypotheses the same type of nonexistence result for nonlinear problems associated with certain systems of parabolic and hyperbolic equations. Author
- Numerical Mathematics