DOUBLY DEGENERATE PARABOLIC EQUATIONS WITH VARIABLE NONLINEARITY I: EXISTENCE OF BOUNDED STRONG SOLUTIONS

摘要

We study the homogeneous Dirichlet problem for the anisotropic parabolic equation with double degeneracy The exponents of nonlinearity m(x, t) > 0, pi(x, t) > 1, and σ(x,t) > 1 are given bounded continuous functions. It is proved that the problem has a bounded solution in a variable-exponent Sobolev space. The main existence result is local in time and is established under minimal restrictions on the low-order terms. It is shown that under further restrictions on b and σ(x, t) the constructed solution can be continued to the arbitrary time interval. The energy estimates are derived.