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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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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 PDF
Trans. Amer. Math. Soc. 347 (1995), 1687-1712 Request permission

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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Additional 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