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On a class of thermal blow-up patterns. (English) Zbl 0969.35520
Summary: We study the patterns of a thermal explosion as described via \(u_t=(\Delta+1)u^m, m>1\). These processes, characterized by an intrinsic length scale, always converge into very simple, universal, space-time separable, axisymmetric pattern(s) with a compact support - referred to as dissipative compactons. When the initial datum is specified on an axisymmetric annulus, though the evolving pattern seems to preserve this symmetry, at a later stage, it collapses very quickly to the center. In a perturbed annulus, local axisymmetric patches of blow-up form instead of a collapse. For a planar, homogeneous, Dirichlet problem, the space-time separability of the emerging pattern is preserved as well, but the competition between the intrinsic and extrinsic characteristic scales generates a wider variety of spatial patterns, with the self-localization taking place on large domains. As the width of the domain diminishes, then depending on the width-length ratio, the emerging pattern first partially, and then fully, attaches to the boundaries. With further decrease of the domain, the emerging separable pattern, instead of exploding, decays algebraically in time.

35K55 Nonlinear parabolic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
Full Text: DOI
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