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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Existence of infinitely many solutions for a forward backward heat equation
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by Klaus Höllig PDF
Trans. Amer. Math. Soc. 278 (1983), 299-316 Request permission

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 \[ {u_t} = \phi {({u_x})_x},\qquad {u_x}(0,t) = {u_x}(1,t) = 0,\qquad u( \cdot ,0) = f,\] 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
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  • 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).
  • O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural′ceva, Lineĭ nye i kvazilineĭ nye uravneniya parabolicheskogo tipa, Izdat. “Nauka”, Moscow, 1967 (Russian). MR 0241821
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Additional Information
  • © Copyright 1983 American Mathematical Society
  • 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