Optimal global second-order regularity and improved integrability for parabolic equations with variable growth

Abstract

Abstract We consider the homogeneous Dirichlet problem for the parabolic equation u t − div ( ∣ ∇ u ∣ p ( x , t ) − 2 ∇ u ) = f ( x , t ) + F ( x , t , u , ∇ u ) {u}_{t}-{\rm{div}}({| \nabla u| }^{p\left(x,t)-2}\nabla u)=f\left(x,t)+F\left(x,t,u,\nabla u) in the cylinder Q T ≔ Ω × ( 0 , T ) {Q}_{T}:= \Omega \times \left(0,T) , where Ω ⊂ R N \Omega \subset {{\mathbb{R}}}^{N} , N ≥ 2 N\ge 2 , is a C 2 {C}^{2} -smooth or convex bounded domain. It is assumed that p ∈ C 0 , 1 ( Q ¯ T ) p\in {C}^{0,1}\left({\overline{Q}}_{T}) is a given function and that the nonlinear source F ( x , t , s , ξ ) F\left(x,t,s,\xi ) has a proper power growth with respect to s s and ξ \xi . It is shown that if p ( x , t ) > 2 ( N + 1 ) N + 2 p\left(x,t)\gt \frac{2\left(N+1)}{N+2} , f ∈ L 2 ( Q T ) f\in {L}^{2}\left({Q}_{T}) , ∣ ∇ u 0 ∣ p ( x , 0 ) ∈ L 1 ( Ω ) {| \nabla {u}_{0}| }^{p\left(x,0)}\in {L}^{1}\left(\Omega ) , then the problem has a solution u ∈ C 0 ( [ 0 , T ] ; L 2 ( Ω ) ) u\in {C}^{0}\left(\left[0,T];\hspace{0.33em}{L}^{2}\left(\Omega )) with ∣ ∇ u ∣ p ( x , t ) ∈ L ∞ ( 0 , T ; L 1 ( Ω ) ) {| \nabla u| }^{p\left(x,t)}\in {L}^{\infty }\left(0,T;\hspace{0.33em}{L}^{1}\left(\Omega )) , u t ∈ L 2 ( Q T ) {u}_{t}\in {L}^{2}\left({Q}_{T}) , obtained as the limit of solutions to the regularized problems in the parabolic Hölder space. The solution possesses the following global regularity properties: ∣ ∇ u ∣ 2 ( p ( x , t ) − 1 ) + r ∈ L 1 ( Q T ) , for any 0 < r < 4 N + 2 , ∣ ∇ u ∣ p ( x , t ) − 2 ∇ u ∈ L 2 ( 0 , T ; W 1 , 2 ( Ω ) ) N . {| \nabla u| }^{2\left(p\left(x,t)-1)+r}\in {L}^{1}\left({Q}_{T}),\hspace{1.0em}\hspace{0.1em}\text{for any\hspace{0.5em}}0\lt r\lt \frac{4}{N+2}\text{}\hspace{0.1em},\hspace{1.0em}{| \nabla u| }^{p\left(x,t)-2}\nabla u\in {L}^{2}{\left(0,T;{W}^{1,2}\left(\Omega ))}^{N}.