Global estimates for solutions of singular parabolic and elliptic equations with variable nonlinearity

We consider the homogeneous Dirichlet problem for the equation u(t) = div ((epsilon(2) + vertical bar del u vertical bar(2))p(x,t) 2/2 del u) + f(x, t), epsilon >= 0, (0.1) in the cylinder Q(T) = Omega x (0, T), Omega subset of R-d, d >= 2, with the variable exponent 2d/d+2 < p(-) <= p(x, t) <= p(+) <= 2, p(+/-) = const. We find sufficient conditions on p, partial derivative Omega, f and u(x, 0) which provide the existence of solutions with the following global regularity properties: u(t) is an element of L-infinity(0, T; L-2(Omega)), vertical bar del u vertical bar is an element of C-0([0, T]; L-2(Omega)), vertical bar u(t)vertical bar(p/p-1), vertical bar u(xixj)vertical bar(p), vertical bar del(ut)vertical bar(p), (epsilon(2) + vertical bar del u vertical bar(2))(p-2/2)vertical bar u(xixj)vertical bar(2) is an element of L-1(Q(T)), p - p(x, t), i, j - 1, 2, ..., d. For the solutions of the stationary counterpart of Eq. (0.1), div (epsilon(2) + vertical bar del v vertical bar(2))(p0(x)-2/2) del v) = Phi(x) in Omega, v = 0 on partial derivative Omega, the inclusions vertical bar v(xixj)vertical bar(p0), (epsilon(2) + vertical bar del v vertical bar(2))(p0-2/2)vertical bar v(xixj)vertical bar(2) is an element of L-1(Omega) are established. (C) 2019 Elsevier Ltd. All rights reserved.