A Qualitative Study of (p, q) Singular Parabolic Equations: Local Existence, Sobolev Regularity and Asymptotic Behavior

The purpose of the article is to study the existence, regularity, stabilization and blow-up results of weak solution to the following parabolic (p, q)-singular equation: {u(t) - Delta(p)u - Delta(q)u = theta u(-delta) + f(x, u), u > 0 in Omega x (0, T), u = 0 on partial derivative Omega x (0, T), u(x, 0) = u(0)(x) in Omega, where Omega is a bounded domain in R-N with C-2 boundary partial derivative Omega, 1 < q < p < infinity, 0 < delta, T > 0, N >= 2 and theta > 0 is a parameter. Moreover, we assume that f : Omega x [0, infinity) -> R is a bounded below Caratheodory function, locally Lipschitz with respect to the second variable uniformly in x is an element of Omega and u(0) is an element of L-infinity(Omega) boolean AND W-0(1,p) (Omega). We distinguish the cases as q-subhomogeneous and q-superhomogeneous depending on the growth of f (hereafter we will drop the term q). In the subhomogeneous case, we prove the existence and uniqueness of the weak solution to problem (P-t) for delta < 2 + 1/p-1. For this, we first study the stationary problems corresponding to (P-t) by using the method of sub- and supersolutions and subsequently employing implicit Euler method, we obtain the existence of a solution to (P-t). Furthermore, in this case, we prove the stabilization result, that is, the solution u(t) of (P-t) converges to u(infinity), the unique solution to the stationary problem, in L-infinity(Omega) as t -> infinity. For the superhomogeneous case, we prove the local existence theorem by taking help of nonlinear semigroup theory. Subsequently, we prove finite time blow-up of solution to problem (P-t) for small parameter theta > 0 in the case delta <= 1 and for all theta > 0 in the case delta > 1. Moreover, we prove higher Sobolev integrability of the solution to purely singular problem corresponding to the steady state of (P-t), which is of independent interest. As a consequence of this, we improve the Sobolev regularity of solution to (P-t) for the case delta < 2 + 1/p-1.