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Singular limit of solutions of $ u\sb t=\Delta u\sp m-A\cdot \nabla (u\sp q/q)$ as $ q\to\infty$

Author: Kin Ming Hui
Journal: Trans. Amer. Math. Soc. 347 (1995), 1687-1712
MSC: Primary 35K55; Secondary 35B40
MathSciNet review: 1290718
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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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Keywords: Asymptotic behaviour, porous medium equation with convection term, existence, uniqueness, nonnegative solutions
Article copyright: © Copyright 1995 American Mathematical Society

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