(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 form… Expand

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

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

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 at… Expand

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 and… Expand