Global existence of classical solutions to a combined chemotaxis–haptotaxis model with logistic source

This paper deals with a mathematical model of cancer invasion of tissue. The model consists of a system of reaction–diffusion-taxis partial differential equations describing interactions between cancer cells, matrix degrading enzymes, and the host tissue. In two space dimensions, we prove global existence and uniqueness of classical solutions to this model for any μ > 0 (where μ is the logistic growth rate of cancer cells). The crucial point of proof is to raise the regularity estimate of a solution from L 1 ( Ω ) to L 3 ( Ω × ( 0 , T ) ) (where Ω ⊂ R 2 is some bounded domain and T > 0 is some constant). This paper develops new estimate techniques and improves greatly our previous results [Y. Tao, M. Wang, Global solution for a chemotactic–haptotactic model of cancer invasion, Nonlinearity 21 (2008) 2221–2238] in 2 dimensions.