Global Existence and Boundedness of Solutions to a Chemotaxis-Haptotaxis Model With Nonlinear Diffusion and Signal Production

In this paper, we investigate the following chemotaxis-haptotaxis model $$\{ {array}{ll} uₜ=∇· (D(u) ∇ u)- ∇· (H(u)∇ v)- ∇· (I(u)∇ w)+ u(a-μ uᵏ⁻¹-λ w),\\ vₜ= v-v+uγ,\\ wₜ=-v w {array}.(*) $$ under homogenous Neumann boundary condition and for a bounded domain $ Ω ⊂ Rⁿ (n≥2)$, with $λ, μ, γ >0$, $k>1$, $a ∈ R$, and $D(u)≥ KD (u+1)ᵐ⁻¹$, $0≤ H(u)≤ χ u(u+1)-α$, $0≤ I(u)≤ ξ u(u+1)-β$ for $KD, χ, ξ>0, m, α, β∈ R$. It has been demonstrated that (i) For $0γ-k+1$ and $β>1-k$, problem ($*$) admits a classical solution $(u, v, w)$ which is globally bounded. (ii) For $2/nγ-k+1/e+1$ and $β>max\{(nγ-2)(nγ+2k-2)/2n-k+1, {(nγ-2)(γ+1/e)}{n}-k+1\}$ or $α>γ-k+1$ and $β>max\{(nγ-2)(nγ+2k-2)/2n-k+1, (nγ-2)(α+k-1)/n-k+1\}$, problem ($*$) admits a classical solution $(u, v, w)$ which is globally bounded.