We consider quasilinear parabolic equations with gradient diffusivity u t ϭdiv(ٌ͉u͉ ٌu)ϩf(u), xR N , tϾ0, where Ϫ1 is a fixed constant and f (u) is a given smooth function. We also study quasilinear parabolic equations with a gradient-dependent coefficient u t ϭh(ٌ͉u͉)⌬uϩ f (u), with a smooth function h(p). For both classes of equations we derive first-order sign-invariants, i.e., firstorder operators preserving their signs on the evolution orbits ͕u(•,t),tϾ0͖. We give a complete description of ͑maximal͒ sign-invariants of prescribed structures. As a consequence, we construct new exact solutions of some quasilinear equations.
We study the asymptotic behaviour as t → ∞ of the solution u = u(x, t) ≧ 0 to the quasilinear heat equation with absorption ut = (um)xx − f(u) posed for t > 0 in a half-line I = { 0 < x < ∞}. For definiteness, we take f(u) = up but the results generalise easily to more general power-like absorption terms f(u). The exponents satisfy m > 1 and p >m. We impose u = 0 on the lateral boundary {x = 0, t > 0}, and consider a non-negative, integrable and compactly supported function uo(x) as initial data. This problem is equivalent to solving the corresponding equation in the whole line with antisymmetric initial data, uo(−x) = −uo(x).
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