The regularity of weak solutions for certain n-dimensional strongly coupled parabolic systems

Abstract

This paper is concerned with the n-dimensional strongly coupled parabolic systems with triangular form in the cylinder Ω×(0,T]\Omega \times (0,T]. We investigate L2{L}^{2} and Hölder regularity of the derivatives of weak solutions (u1,u2)\left({u}_{1},{u}_{2}) for the systems in the following two cases: one is that the boundedness of u1{u}_{1} and u2{u}_{2} has not been shown in existence result of solutions; the other is that the boundedness of u1{u}_{1} or u2{u}_{2} has been shown in existence result of solutions. By using difference ratios and Steklov averages methods and various estimates, we prove that if (u1,u2)\left({u}_{1},{u}_{2}) is a weak solution of the system, then for any Ω′⊂⊂Ω\Omega ^{\prime} \subset \hspace{-0.3em}\subset \hspace{0.33em}\Omega and t′∈(0,T)t^{\prime} \in \left(0,T), u1,u2{u}_{1},{u}_{2} belong to Cα′,α′/2(Ω¯′×[t′,T]){C}^{\alpha ^{\prime} ,\alpha ^{\prime} \text{/}2}\left(\bar{\Omega }^{\prime} \times \left[t^{\prime} ,T]) and W22,1(Ω′×(t′,T]){W}_{2}^{2,1}\left(\Omega ^{\prime} \times (t^{\prime} ,T]) under certain conditions, and u1,u2{u}_{1},{u}_{2} belong to C2+α′,1+α′/2(Ω¯′×[t′,T]){C}^{2+\alpha ^{\prime} ,1+\alpha ^{\prime} \text{/}2}\left(\bar{\Omega }^{\prime} \times \left[t^{\prime} ,T]) under stronger assumptions. Applications of these results are given to two ecological models with cross-diffusion.

Keywords

• strongly coupled parabolic systems
• regularity
• weak solutions
• triangular systems
• 35k55
• 35b65
• 35k57