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Uniqueness and stability of regional blow-up in a porous-medium equation. (English) Zbl 1018.35062

Summary: We study the blow-up phenomenon for the porous-medium equation in \(\mathbb{R}^N\), \(N\geq 1\), \[ u_t=\Delta u^m+u^m, \] \(m>1\), for nonnegative, compactly supported initial data. A solution \(u(x,t)\) to this problem blows-up at a finite time \(\overline T>0\). Our main result asserts that there is a finite number of points \(x_1,\dots, x_k\in\mathbb{R}^N\), with \(|x_i-x_j |\geq 2R^*\) for \(i\neq j\), such that \[ \lim_{t\to\overline T}(\overline T-t)^{1\over m-1} u(t,x)=\sum^k_{j=1} w_*\bigl(|x-x_j|\bigr). \] Here \(w_* (|x|)\) is the unique nontrivial, nonnegative compactly supported, radially symmetric solution of the equation \(\Delta w^m+w^m- {1\over m-1}w=0\) in \(\mathbb{R}^N\) and \(R^*\) is the radius of its support. Moreover \(u(x,t)\) remains uniformly bounded up to its blow-up time on compact subsets of \(\mathbb{R}^N \setminus \bigcup^k_{j=1} \overline B(x_j,R^*)\). The question becomes reduced to that of proving that the \(\omega\)-limit set in the problem \(v_t=\Delta v^m+v^m-{1\over m-1}v\) consists of a single point when its initial condition is nonnegative and compactly supported.

MSC:

35Q35 PDEs in connection with fluid mechanics
35K65 Degenerate parabolic equations
76S05 Flows in porous media; filtration; seepage
35B40 Asymptotic behavior of solutions to PDEs
35K57 Reaction-diffusion equations
35B35 Stability in context of PDEs
35C20 Asymptotic expansions of solutions to PDEs
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