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Stabilization of solutions of a degenerate nonlinear diffusion problem
(I) [ 4 = (u% + f(u) in (-L,L) X R+, u(tL,t)=O in R+, u(x, 0) = uo(x) in [-L,L], where m > 1 is a parameter, f is locally Lipschitz continuous, f(0) = 0, and u. is bounded. Problems of this formExpand
Large time behaviour of solutions of the porous medium equation in bounded domains
If m = 1 the partial differential equation in Problem (I) is just the classical equation of heat conduction and it is well known that under appropriate conditions on u,, and 0, u( ., t) + 0 as t -+Expand
A New Class of Entropy Solutions of the Buckley-Leverett Equation
This extension of the Buckley–Leverett (BL) equation including a third order mixed derivatives term and models the dynamic effects in the pressure difference between the two phases leads to admissible shocks for the original BL equation, which violate the Oleinik entropy condition and are therefore called nonclassical. Expand
A very singular solution of the heat equation with absorption
Consider the Cauchy problem ut -du + u p ----0 on RN• (0, oo) (I.1) u > 0 on RN• (0, oo) (1.2) u(X, O) = c ~(x) on R, N, (1.3) where N _--> 1, c > 0 is a constant and ~(x) denotes the Dirac mass atExpand
Saddle solutions of the bistable diffusion equation
Stationary solutions of the bistable Cahn-Allen diffusion equation in the plane are constructed, which are positive in quadrants 1 and 3 and negative in the other two quadrants. They are unique andExpand
Spatial Patterns: Higher Order Models in Physics and Mechanics
Preface Outline 1. Introduction Part I: The Symmetric Bistable Equation 2. Real Eigenvalues 3. Estimates 4. Periodic Solutions 5. Kinks and Pulses 6. Chaotic Solutions 7. Variational Problems PartExpand