Some nonexistence theorems for initial-boundary value problems with nonlinear boundary constraints

Levine, H. A.; Payne, L. E.

doi:10.1090/S0002-9939-1974-0364841-6

Some nonexistence theorems for initial-boundary value problems with nonlinear boundary constraints

Authors: H. A. Levine and L. E. Payne
Journal: Proc. Amer. Math. Soc. 46 (1974), 277-284
MSC: Primary 35G30
DOI: https://doi.org/10.1090/S0002-9939-1974-0364841-6
MathSciNet review: 0364841
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Abstract: The present work extends previous results of the authors [4] on nonexistence of solutions of classes of second order equations subject to certain nonlinear boundary constraints and appropriate hypotheses on the data. Similar results are derived here for more general classes of higher order equations.

[1] R. A. Knops, H. A. Levine and L. E. Payne, Nonexistence, instability and growth theorems for solutions to an abstract nonlinear equation with applications in elastodynamics Arch. Rational Mech. Anal. (to appear).
[2] Howard A. Levine, Some nonexistence and instability theorems for solutions of formally parabolic equations of the form 𝑃𝑢_{𝑡}=-𝐴𝑢+ℱ(𝓊), Arch. Rational Mech. Anal. 51 (1973), 371–386. MR 348216, https://doi.org/10.1007/BF00263041
[3] Howard A. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form 𝑃𝑢_{𝑡𝑡}=-𝐴𝑢+\cal𝐹(𝑢), Trans. Amer. Math. Soc. 192 (1974), 1–21. MR 344697, https://doi.org/10.1090/S0002-9947-1974-0344697-2
[4] Howard A. Levine and Lawrence E. Payne, Nonexistence theorems for the heat equation with nonlinear boundary conditions and for the porous medium equation backward in time, J. Differential Equations 16 (1974), 319–334. MR 470481, https://doi.org/10.1016/0022-0396(74)90018-7