Blow-Up Phenomena of a Cancer Invasion Model with Nonlinear Diffusion and Haptotaxis Term
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Blow-Up Phenomena of a Cancer Invasion Model with Nonlinear Diffusion and Haptotaxis Term L. Shangerganesh1 · G. Sathishkumar2 · N. Nyamoradi3 · S. Karthikeyan2 Received: 17 February 2020 / Revised: 11 July 2020 / Accepted: 7 August 2020 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2020
Abstract In this paper, we consider a nonlinear cancer invasion mathematical model with proliferation, growth and haptotaxis effects. We obtain lower bounds for the finite-time blow-up of solutions of the considered system with nonlinear diffusion operator when blow-up occurs. We have assumed both the Dirichlet and Neumann boundary conditions in Rn , n ≥ 2 to attain the desire result. Keywords Blow-up · Lower bound · Cancer invasion · Reaction–diffusion · Haptotaxis Mathematics Subject Classification 35B44 · 34L15 · 35K57 · 92C17
1 Introduction This paper investigates the blow-up of the following cancer invasion mathematical model with nonlinear diffusion operator:
Communicated by Yong Zhou.
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N. Nyamoradi [email protected]; [email protected] L. Shangerganesh [email protected] G. Sathishkumar [email protected] S. Karthikeyan [email protected]
1
Department of Applied Sciences, National Institute of Technology, Goa 403 401, India
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Department of Mathematics, Periyar University, Salem 636 011, India
3
Department of Mathematics, Faculty of Sciences, Razi University, 67149 Kermanshah, Iran
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L. Shangerganesh et al.
⎫ ∂u ⎪ = ∇ · (φ(u)∇u) − ∇ · (χ u∇v) + μu(1 − u − v), x ∈ , t > 0, ⎪ ⎪ ⎪ ∂t ⎬ ∂v = −σ vw + ρv(1 − u − v), x ∈ , t > 0, (1.1) ⎪ ∂t ⎪ ⎪ ∂w ⎪ = dw + ζ u(1 − w) − τ w, x ∈ , t > 0, ⎭ ∂t with the following initial and boundary conditions: ∂u ∂v ∂w + β1 u = 0, α2 + β2 v = 0 and α3 + β3 w = 0, x ∈ ∂, t > 0, ∂ν ∂ν ∂ν u(x, 0) = u 0 (x) ≥ 0, v(x, 0) = v0 (x) ≥ 0 and w(x, 0) = w0 (x) ≥ 0, x ∈ ,
α1
(1.2) where ⊂ Rn (n ≥ 2) is a bounded domain with smooth boundary ∂. Here u = u(x, t), v = v(x, t) and w = w(x, t) represents the cancer cell density, extra cellular matrix density and matrix degrading enzymes concentration respectively. Here, u 0 (x), v0 (x) and w0 (x) are non-negative smooth initial functions of u, v and w respectively. Further, χ , μ, σ, ρ, d, ζ, τ, αi (i = 1, 2, 3) and βi (i = 1, 2, 3) are non-negative constants. Furthermore, we assume that the diffusion coefficient φ(u) satisfies φ(u) ≥ a1 + a2 u 2 with some positive constants a1 and a2 . A quite large amount of works have been done by many researchers on the cancer invasion mathematical models in different physical circumstances, for example, see [1,4,21,33,35] and references therein. In particular, for solvability of cancer dynamics models, we refer to the papers [7,9,12,15,23–25,27,32] and references cited. Blow-up represents an uncontrolled growth of damaged cells and the patient’s health condition that become critical when blow-up occurs and finding the lower bounds for blow-up time are very important. Chen and Yu [5] discussed the global existence and blow-up property, which
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