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We consider the homogeneous Dirichlet problem for the parabolic equation ut−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 QT≔Ω×(0,T){Q}_{T}:= \Omega \times \left(0,T), where Ω⊂RN\Omega \subset {{\mathbb{R}}}^{N}, N≥2N\ge 2, is a C2{C}^{2}-smooth or convex bounded domain. It is assumed that p∈C0,1(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 ss and ξ\xi . It is shown that if p(x,t)>2(N+1)N+2p\left(x,t)\gt \frac{2\left(N+1)}{N+2}, f∈L2(QT)f\in {L}^{2}\left({Q}_{T}), ∣∇u0∣p(x,0)∈L1(Ω){| \nabla {u}_{0}| }^{p\left(x,0)}\in {L}^{1}\left(\Omega ), then the problem has a solution u∈C0([0,T];L2(Ω))u\in {C}^{0}\left(\left[0,T];\hspace{0.33em}{L}^{2}\left(\Omega )) with ∣∇u∣p(x,t)∈L∞(0,T;L1(Ω)){| \nabla u| }^{p\left(x,t)}\in {L}^{\infty }\left(0,T;\hspace{0.33em}{L}^{1}\left(\Omega )), ut∈L2(QT){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∈L1(QT),for any 0 |
