The Cauchy problem for a degenerate parabolic equation with inhomogeneous density and source in the class of slowly decaying initial data (Q2901880)

The authors study the following Cauchy problem NEWLINE\[NEWLINE \begin{aligned} & \rho(x)u_t = \mathrm{div}\left(u^{m-1}|Du|^{\lambda - 1}Du\right) + \rho(x)u^p,\\ & u(x,0) = u_0(x), \quad x \in \mathbb R^N, \end{aligned} NEWLINE\]NEWLINE where \((x,t) \in Q_T = \mathbb R^N\times (0,T)\), \(T > 0\), \(x \in \mathbb R^N\). It is assumed that \(1 < \lambda + 1 < N\), \(m + \lambda - 2 > 0\), \(\rho(x) = |x|^{-l}\), \(0 \leq l < \lambda + 1\) and \(u_0(x)\) is a measurable non-negative function in \(L^1_{\rho,loc}(\mathbb R^N)\) such that the following weighted norm is finite for some \(\alpha \in (0,N - l)\) and \(\theta = \max\{1,l(N - \lambda - 1)/(\alpha(\lambda + 1))\}\): NEWLINE\[NEWLINE |||u_0|||_{\rho,\theta} = \sup_{x_0\in \mathbb R^N}(E_{B_{d(|x_0|)}(x_0)}\rho u_0^{\theta})^{1/\theta} < \infty, NEWLINE\]NEWLINE where NEWLINE\[NEWLINE E_{B_{d(|x_0|)}(x_0)}\rho u_0^{\theta} = \frac{\int_{B_{d(|x_0|)}(x_0)}\rho u^{\theta}_0(x)dx} {\int_{B_{d(|x_0|)}(x_0)}\rho dx}. NEWLINE\]NEWLINE Here \(B_{R}(x_0) = \{x \in \mathbb R^N\,:\, |x - x_0| \leq R\}\) and \(d(a) = (1 + a)^{(\alpha\theta + l)/N}\), \(a \geq 0\).NEWLINENEWLINEThe aim of the paper to find conditions for the existence and non-existence of global in time solutions of this problem in a class of initial functions that need not belong \(L_{\rho}^1(\mathbb R^N)\). In the case when global in time solutions exist, the authors obtain exact a priory estimates for large values of time. Notice that even the simplest version of the equation (\(m = 1\), \(\lambda = 1\) and \(\rho(x) \equiv 1\)) admits solutions which become unbounded in finite time.NEWLINENEWLINEThe main results of the paper reads: Suppose that \(p > p^*_{\alpha}(l) = m + \lambda + 1 - l)/\alpha\) and NEWLINE\[NEWLINE |||u_0|||_{\rho,\theta} + \|u_0\|_{L^q(\mathbb R^N)} \leq \delta NEWLINE\]NEWLINE where \(\delta > Q = (N -l)(p - m - \lambda + 1)/(\lambda + 1 - l)\) and \(\delta > 0\) is a sufficiently small number depending only on the parameters of the above-mentioned problem. Then the problem has a global in time solution and the following estimates holds for every \(t \in (0,\infty)\): NEWLINE\[NEWLINE \|u(\cdot,t)\|_\infty\leq \gamma_1t^{-\frac{N - l}{H_{l,\theta}}}(\gamma_2 + (t|||u_0|||_{\rho,\theta}^{m + \lambda - 2})^{\frac{N - \alpha\theta - l}{G_{\alpha}}}) ^{\frac{\lambda + 1 - \lambda}{H_{l,\theta}}}|||u_0|||^{\frac{(\lambda + 1 - l)\theta} {H_{l,\theta}}}, NEWLINE\]NEWLINE where NEWLINE\[NEWLINE H_{l,\theta} = (N - l)(m + \lambda - 2) + (\lambda + 1 - l)\theta, \quad G_{\alpha} = \alpha(m + \lambda - 2) + \lambda + 1 - l. NEWLINE\]NEWLINENEWLINENEWLINESuppose that \(p < p^*_{\alpha}\), \(u(x,t)\) is a solution of the above-mentioned problem and the initial function \(u_0(x)\) satisfies the condition NEWLINE\[NEWLINE (R + |x_0|)^{\alpha(1 - \theta)}E_{B_{d(|x_0|)}(x_0)}\rho u_0^{1 - \theta} \geq \gamma_1 NEWLINE\]NEWLINE for some \(x_0 \in \mathbb R^N\), \(\theta \in (0,1)\) and arbitrary \(R > 0\). Then \(u(x,t)\) blows up at a finite time, that is, there are \(0 < R_1 < \infty\), and \(T\), \(0 < T < \infty\), such that NEWLINE\[NEWLINE \int_{B_{R_1(x_0)}}\rho u^{1 - \theta}(x,t)dx \to \infty \quad \text{as } t \to T. NEWLINE\]