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

Abstract

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 L 2 {L}^{2} and Hölder regularity of the derivatives of weak solutions ( u 1 , u 2 ) \left({u}_{1},{u}_{2}) for the systems in the following two cases: one is that the boundedness of u 1 {u}_{1} and u 2 {u}_{2} has not been shown in existence result of solutions; the other is that the boundedness of u 1 {u}_{1} or u 2 {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 ( u 1 , u 2 ) \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) , u 1 , u 2 {u}_{1},{u}_{2} belong to C α ′ , α ′ / 2 ( Ω ¯ ′ × [ t ′ , T ] ) {C}^{\alpha ^{\prime} ,\alpha ^{\prime} \text{/}2}\left(\bar{\Omega }^{\prime} \times \left[t^{\prime} ,T]) and W 2 2 , 1 ( Ω ′ × ( t ′ , T ] ) {W}_{2}^{2,1}\left(\Omega ^{\prime} \times (t^{\prime} ,T]) under certain conditions, and u 1 , u 2 {u}_{1},{u}_{2} belong to C 2 + 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.