The porous media equation with nonconstant coefficients (Q5954412)

There is proved existence of generalized nonnegative solutions and obtained some results about asymptotic behaviour as \(t\to\infty\) of mentioned solutions to the following Cauchy problem: NEWLINE\[NEWLINE (P)\left\{\begin{aligned} & \frac{\partial u}{\partial t} = \frac{\partial}{\partial x} \Big(A(x,t)\frac{\partial u^m}{\partial x}\Big) \text{ in } Q=\mathbb{R}\times\mathbb{R}^+, \;m>1, \\ & u(x,0) = u_0(x)\in L^\infty(\mathbb{R})\cap C(\mathbb{R})\cap L^1(\mathbb{R}), \;u_0(x)\geq 0\text{ in }\mathbb{R}. \end{aligned}\right. NEWLINE\]NEWLINE Theorem 1 (concentration of mass). Let additionally \(A\in C^{2,2}(\overline Q)\) and \(u(x,t)\) be the solution of \((P)\). Then for any \(\varepsilon>0\) there exist \(a>0\) and \(b>0\) such that NEWLINE\[NEWLINE \begin{aligned} & \text{if } x\geq\eta t^{\frac{1}{m+1}}+a,\text{ then } v(x,t) \equiv\int^x_{-\infty} u(s,t) ds\geq M-\varepsilon, \\ & \text{if } x\leq-\eta t^{\frac{1}{m+1}}-b, \text{ then }v(x,t)\leq\varepsilon, \end{aligned} NEWLINE\]NEWLINE where \(\eta\) is a constant which does not depend on \(\varepsilon, M=\|u_0\|_{L^1(\mathbb{R})}\). NEWLINENEWLINENEWLINETheorem 2 (expanding of mass). Let \(u\) and \(v\) be the same as in Theorem 1. Then for any \(\varepsilon >0\) there exist \(a>0, b>0\) such that NEWLINE\[NEWLINE \begin{aligned} & \text{if }x\leq\eta_1 t^{\frac{1}{m+1}}-b,\text{ then }v(x,t) \leq M-\varepsilon, \\ & \text{if }x\geq-\eta_1 t^{\frac{1}{m+1}}+a, \text{ then }v(x,t)\geq\varepsilon, \end{aligned} NEWLINE\]NEWLINE where \(\eta_1\) does not depend on \(\varepsilon\). NEWLINENEWLINENEWLINETheorem 3 (bound from below). In assumptions of Theorem 1 there exist positive constants \(c,\eta\) and \(T\) such that for \(t\geq T\): NEWLINE\[NEWLINE u(x,t)\geq C t^{-\frac{1}{m+1}} \text{ if }|x|\leq \eta t^{\frac{1}{m+1}}. NEWLINE\]NEWLINE There is studied the large time behaviour of solution under additional assumption on \(A\), motivated by the study of special quasilinear system. The case of non-smooth \(A(x,t)\) is considered too.