On a class of nonlocal evolution equations with the p[∇u]-Laplace operator

We study the homogeneous Dirichlet problem for the class of singular parabolic equationsut−div(|∇u|p[∇u]−2∇u)=fin Ω×(0,T), where Ω⊂Rd, d≥2, is a smooth domain. The exponent p nonlocally depends on the gradient of the solution: p is a given function defined byp[∇u]≡p(l(|∇u|)),l(|s|)=∫Ω|s|αdx with a constant α∈(1,2]. We find sufficient conditions on the data that guarantee global in time existence and uniqueness of a strong solution of the problem. It is shown that the problem has a solution if either u0 and f, or p′(s) are sufficiently small.