On the localization of solutions of doubly nonlinear parabolic equations with nonstandard growth in filtration theory

We study the properties of space localization of weak solutions of the equation. partial derivative(t) (vertical bar u vertical bar(m(x,t)) sign u) - div A(x,t,u,del u) + C(x,t,u) = 0 which appears in the mathematical description of filtration of an ideal barotropic gas in a porous medium. The functionsAand C are assumed to satisfy the nonstandard growth conditions: for all(x, t) is an element of Omega x (0, T), s is an element of R, xi is an element of R-n vertical bar A(x, t, s, xi)vertical bar <= a(1) vertical bar xi vertical bar(p(x,) (t)-1), A(x, t, s, xi) . xi >= a(0)vertical bar xi vertical bar(p(x,) (t)), C(x, t, s)s >= c(0)vertical bar s vertical bar(sigma(x, t)) - f(x, t) with some positive constants a(0), a(1), c(0) and measurable bounded functions p(x, t) > 1, sigma(x, t) > 1, m( x, t) > 0. It is shown that if ess sup m(x, t) + 1 < ess inf p(x, t), f(x, t) = 0, and m(x, t), p(x, t) meet certain regularity requirements, then every weak solution possesses the property of finite speed of propagation of disturbances from the initial data. In the case that u(x, 0) = 0 in a ball B-r(x(0)) and f(x, t) = 0 in Br (x(0)) x (0, T), the solutions display the waiting time property: if parallel to vertical bar u(0)vertical bar(m+1)parallel to(1), (B rho(x0)) + parallel to vertical bar f vertical bar(m+1/m)parallel to(1, Q rho(x0)) <= is an element of(rho - rho(0))(+)(1/(1-nu)) with a positive exponent nu, depending on m(x, t) and p(x, t), and a sufficiently small epsilon> 0, then there exists t* > 0 such that u = 0 in B-r(x(0)) x (0, t*).