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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Singular limit of solutions of $u_ t=\Delta u^ m-A\cdot \nabla (u^ q/q)$ as $q\to \infty$
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by Kin Ming Hui
Trans. Amer. Math. Soc. 347 (1995), 1687-1712
DOI: https://doi.org/10.1090/S0002-9947-1995-1290718-6

Abstract:

We will show that the solutions of ${u_t} = \Delta {u^m} - A\nabla ({u^q}/q)$ in ${R^n} \times (0,T),T > 0,m > 1,u(x,0) = f(x) \in {L^1}({R^n}) \cap {L^\infty }({R^n})$ converge weakly in ${({L^\infty }(G))^ * }$ for any compact subset $G$ of ${R^n} \times (0,T)$ as $q \to \infty$ to the solution of the porous medium equation ${\upsilon _t} = \Delta {\upsilon ^m}$ in ${R^n} \times (0,T)$ with $\upsilon (x,0) = g(x)$ where $g \in {L^1}({R^n}),0 \leqslant g \leqslant 1$, satisfies $g(x) + {(g(x))_{{x_1}}} = f(x)\quad {\text {in}}\quad \mathcal {D}’\left ( {{R^n}} \right )$ for some function $\tilde {g}(x) \in {L^1}({R^n}),\quad \tilde {g}(x) \geqslant 0$ such that $g(x) = f(x),\quad \tilde {g}(x) = 0$ whenever $g(x) < 1$ a.e. $x \in {R^n}$. The convergence is uniform on compact subsets of ${R^n} \times (0,T)\quad {\text {if}}\quad f \in {C_0}({R^n})$.
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Bibliographic Information
  • © Copyright 1995 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 347 (1995), 1687-1712
  • MSC: Primary 35K55; Secondary 35B40
  • DOI: https://doi.org/10.1090/S0002-9947-1995-1290718-6
  • MathSciNet review: 1290718