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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

Existence of infinitely many solutions for a forward backward heat equation


Author: Klaus Höllig
Journal: Trans. Amer. Math. Soc. 278 (1983), 299-316
MSC: Primary 35K60
MathSciNet review: 697076
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Abstract: Let $ \phi $ be a piecewise linear function which satisfies the condition $ s\phi (s) \geqslant c{s^2},c > 0,s \in {\mathbf{R}}$, and which is monotone decreasing on an interval $ (a,b) \subset {{\mathbf{R}}_ + }$. It is shown that for $ f \in {C^2}[0,1]$, with $ \max f^\prime > a$, there exists a $ T > 0$ such that the initial boundary value problem

$\displaystyle {u_t} = \phi \,{({u_x})_x},\qquad {u_x}(0,t) = {u_x}(1,t) = 0,\qquad u( \cdot ,0) = f,$

has infinitely many solutions $ u$ satisfying $ \parallel \;u\;{\parallel_{\alpha }},\parallel \;{u_x}{\parallel_{\infty }},\parallel \;{u_t}{\parallel_{2}} \leqslant c(f,\phi )$ on $ [0,1] \times [0,T]$.

References [Enhancements On Off] (What's this?)

  • [BNW] J. Bona, J. Nohel and L. Wahlbin, Private communication.
  • [D] William Alan Day, The thermodynamics of simple materials with fading memory, Springer-Verlag, New York, 1972. Springer Tracts in Natural Philosophy, Vol. 22. MR 0366234 (51 #2482)
  • [HN] K. Höllig and J. A. Nohel, A diffusion equation with a nonmonotone constitutive function, NATO/London Math. Soc. Conference on Systems of Partial Differential Equations, Oxford, 1982 (to appear).
  • [L] O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural′ceva, Linear and quasilinear equations of parabolic type, Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I., 1968 (Russian). MR 0241822 (39 #3159b)
  • [SA] G. Strang and M. Abdel-Naby, Lecture Notes in Engineering, Springer-Verlag, Berlin and New York (to appear).

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1983-0697076-8
PII: S 0002-9947(1983)0697076-8
Keywords: Parabolic equation, nonlinear, diffusion, nonmonotone constitutive function, existence, nonuniqueness
Article copyright: © Copyright 1983 American Mathematical Society