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

Abstract

Abstract 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 ) {(p,q)} -singular equation: ($\mathrm{P}_{t}$) { u t - Δ p ⁢ u - Δ q ⁢ u = ϑ ⁢ u - δ + f ⁢ ( x , u ) , u > 0 in  ⁢ Ω × ( 0 , T ) , u = 0 on  ⁢ ∂ ⁡ Ω × ( 0 , T ) , u ⁢ ( x , 0 ) = u 0 ⁢ ( x ) in  ⁢ Ω , \displaystyle{}\left\{\begin{aligned} \displaystyle{}u_{t}-\Delta_{p}u-\Delta_% {q}u&\displaystyle=\vartheta u^{-\delta}+f(x,u),\quad u>0&&\displaystyle% \phantom{}\text{in }\Omega\times(0,T),\\ \displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{on }\partial% \Omega\times(0,T),\\ \displaystyle u(x,0)&\displaystyle=u_{0}(x)&&\displaystyle\phantom{}\text{in }% \Omega,\end{aligned}\right. where Ω is a bounded domain in ℝ N {\mathbb{R}^{N}} with C 2 {C^{2}} boundary ∂ ⁡ Ω {\partial\Omega} , 1 < q < p < ∞ {1<q<p<\infty} , 0 < δ , T > 0 {0<\delta,T>0} , N ≥ 2 {N\geq 2} and ϑ > 0 {\vartheta>0} is a parameter. Moreover, we assume that f : Ω × [ 0 , ∞ ) → ℝ {f:\Omega\times[0,\infty)\to\mathbb{R}} is a bounded below Carathéodory function, locally Lipschitz with respect to the second variable uniformly in x ∈ Ω {x\in\Omega} and u 0 ∈ L ∞ ⁢ ( Ω ) ∩ W 0 1 , p ⁢ ( Ω ) {u_{0}\in L^{\infty}(\Omega)\cap W^{1,p}_{0}(\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 {\mathrm{P}_{t}} ) for δ < 2 + 1 p - 1 {\delta<2+\frac{1}{p-1}} . For this, we first study the stationary problems corresponding to ( P t {\mathrm{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 {\mathrm{P}_{t}} ). Furthermore, in this case, we prove the stabilization result, that is, the solution u ⁢ ( t ) {u(t)} of ( P t {\mathrm{P}_{t}} ) converges to u ∞ {u_{\infty}} , the unique solution to the stationary problem, in L ∞ ⁢ ( Ω ) {L^{\infty}(\Omega)} as t → ∞ {t\rightarrow\infty} . 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 {\mathrm{P}_{t}} ) for small parameter ϑ > 0 {\vartheta>0} in the case δ ≤ 1 {\delta\leq 1} and for all ϑ > 0 {\vartheta>0} in the case δ > 1 {\delta>1} . Moreover, we prove higher Sobolev integrability of the solution to purely singular problem corresponding to the steady state of ( P t {\mathrm{P}_{t}} ), which is of independent interest. As a consequence of this, we improve the Sobolev regularity of solution to ( P t {\mathrm{P}_{t}} ) for the case δ < 2 + 1 p - 1 {\delta<2+\frac{1}{p-1}} .