Double-phase parabolic equations with variable growth and nonlinear sources

Abstract

We study the homogeneous Dirichlet problem for the parabolic equations ut−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 FF. The initial function belongs to a Musielak-Orlicz space defined by the flux. The functions aa, pp, and qq are Lipschitz-continuous, a(z)a\left(z) is nonnegative, and may vanish on a set of nonzero measure. The exponents pp, and qq satisfy the balance conditions 2NN+20\varepsilon \gt 0.

Keywords

• singular and degenerate parabolic equation
• double phase problem
• variable nonlinearity
• nonlinear source
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