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Transactions of the American Mathematical Society

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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
DOI: https://doi.org/10.1090/S0002-9947-1983-0697076-8
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?)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1983-0697076-8
Keywords: Parabolic equation, nonlinear, diffusion, nonmonotone constitutive function, existence, nonuniqueness
Article copyright: © Copyright 1983 American Mathematical Society

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