Double-phase parabolic equations with variable growth and nonlinear sources

Abstract

Abstract We study the homogeneous Dirichlet problem for the parabolic equations u t − div ( A ( z , ∣ ∇ u ∣ ) ∇ u ) = F ( z , u , ∇ u ) , z = ( x , t ) ∈ Ω × ( 0 , T ) , {u}_{t}-{\rm{div}}\left({\mathcal{A}}\left(z,| \nabla u| )\nabla u)=F\left(z,u,\nabla u),\hspace{1.0em}z=\left(x,t)\in \Omega \times \left(0,T), with the double phase flux A ( z , ∣ ∇ u ∣ ) ∇ u = ( ∣ ∇ u ∣ p ( z ) − 2 + a ( z ) ∣ ∇ u ∣ q ( z ) − 2 ) ∇ u {\mathcal{A}}\left(z,| \nabla u| )\nabla u=(| \nabla u{| }^{p\left(z)-2}+a\left(z)| \nabla u{| }^{q\left(z)-2})\nabla u and the nonlinear source F F . The initial function belongs to a Musielak-Orlicz space defined by the flux. The functions a a , p p , and q q are Lipschitz-continuous, a ( z ) a\left(z) is nonnegative, and may vanish on a set of nonzero measure. The exponents p p , and q q satisfy the balance conditions 2 N N + 2 < p − ≤ p ( z ) ≤ q ( z ) < p ( z ) + r ∗ 2 \frac{2N}{N+2}\lt {p}^{-}\le p\left(z)\le q\left(z)\lt p\left(z)+\frac{{r}^{\ast }}{2} with r ∗ = r ∗ ( p − , N ) {r}^{\ast }={r}^{\ast }\left({p}^{-},N) , p − = min Q ¯ T p ( z ) {p}^{-}={\min }_{{\overline{Q}}_{T}}\hspace{0.33em}p\left(z) . It is shown that under suitable conditions on the growth of F ( z , u , ∇ u ) F\left(z,u,\nabla u) with respect to the second and third arguments, the problem has a solution u u with the following properties: u t ∈ L 2 ( Q T ) , ∣ ∇ u ∣ p ( z ) + δ ∈ L 1 ( Q T ) for every 0 ≤ δ < r ∗ , ∣ ∇ u ∣ s ( z ) , a ( z ) ∣ ∇ u ∣ q ( z ) ∈ L ∞ ( 0 , T ; L 1 ( Ω ) ) with s ( z ) = max { 2 , p ( z ) } . \begin{array}{l}{u}_{t}\in {L}^{2}\left({Q}_{T}),\hspace{1.0em}| \nabla u{| }^{p\left(z)+\delta }\in {L}^{1}\left({Q}_{T})\hspace{1.0em}\hspace{0.1em}\text{for every}\hspace{0.1em}\hspace{0.33em}0\le \delta \lt {r}^{\ast },\\ | \nabla u{| }^{s\left(z)},\hspace{0.33em}a\left(z)| \nabla u{| }^{q\left(z)}\in {L}^{\infty }\left(0,T;\hspace{0.33em}{L}^{1}\left(\Omega ))\hspace{1em}{\rm{with}}\hspace{0.33em}s\left(z)=\max \left\{2,p\left(z)\right\}.\end{array} Uniqueness is proven under stronger assumptions on the source F F . The same results are established for the equations with the regularized flux A ( z , ( ε 2 + ∣ ∇ u ∣ 2 ) 1 / 2 ) ∇ u {\mathcal{A}}(z,{({\varepsilon }^{2}+| \nabla u{| }^{2})}^{1\text{/}2})\nabla u , ε > 0 \varepsilon \gt 0 .