Lower bounds for the finite-time blow-up of solutions of a cancer invasion model

In this article, we consider non-negative solutions of the nonlinear cancer invasion mathematical model involving proliferation and growth functions with homogeneous Neumann and Robin type boundary conditions. We first obtain lower bounds for the finite time blow-up of solutions in R3 with assumed boundary conditions. Finally, we extend the blow-up results of the given system in R2 using first-order differential inequality techniques and under appropriate assumptions on data.


Introduction
Cancer is the most threatening disease to the society due its mortality rate among affected patients.In the past few years, many works presented for the acid-mediated invasion hypothesis and it is proposing that tumour acidification confers an advantage to the tumor cells by producing a harsh environment.Further this process facilitates invasion of tumor cells into the normal cells by producing matrix degrading enzymes.Partial differential equation (PDE) is one of the best modelling tool to study acid mediated cancer dynamics.PDEs have been used for many cancer invasion mathematical models, for example, see [2,4,5,7,10,15,[24][25][26][27][28] and the references therein.This paper investigate the properties of non-negative solutions of the following nonlinear coupled cancer invasion mathematical model in a smooth bounded domain Ω ⊂ R N , N = 2, 3 : in Ω × (0, I), in Ω × (0, I), w t − d 2 ∆w = ζu(1 − w) − νw in Ω × (0, I), ∂u ∂n + h 1 (t)u = 0, ∂v ∂n = 0, ∂w ∂n + h 2 (t)w = 0 on ∂Ω × (0, I), u = u 0 (x), v = v 0 (x), w = w 0 (x) in Ω.
Corresponding author.Email: gskmathspu@gmail.com The mathematical model consists of three unknown variables namely cancer cell density u(x, t), extra cellular matrix (ECM) density v(x, t) and matrix degrading enzymes (MDE) concentration w(x, t).The proliferation rate of cancer cells are assumed to have a logistic growth and is given by µu(1 − u − v).Here µ > 0 is a growth rate constant.Further, MDEs produced by the cancer cells degrade most of the components of ECM.Here the degradation processes is modeled by kvw, k is a positive constant.We also assumed that the remodeling growth of ECM follows a logistic growth, that is, ρv(1 − u − v), where ρ is a positive constant.Moreover the decay and growth rates of MDEs are respectively modeled by νw and ζu(1 − w), where ν, ζ are positive constants.Further, d 1 and d 2 are positive constant diffusion coefficients of cancer cell density and MDE concentration respectively.Finally, u 0 (x), v 0 (x) and w 0 (x) are non-negative functions and represent the initial conditions of u, v and w respectively.Here, we have considered the natural boundary conditions for u, v and w where h i (t), i = 1, 2 are non-negative functions.
Due to the wide range of applications of nonlinear parabolic partial differential equations in many branches of engineering, physics, biology and other sciences, the study of nonlinear parabolic system has became an important field in mathematical analysis.In particular, the study of blow-up for nonlinear parabolic systems received much attention in the last few decades, for instance, see [1,3,6,8,17,21,29] and the references cited in these papers.In the above mentioned papers, various methods were developed and used to study the global existence of solutions, blow-up of solutions, asymptotic behaviours of solutions, upper bound and lower bounds for finite time blow-up of solutions.We refer the interested readers to [9, 13, 14, 18-20, 22, 23] and and the references therein.
Existence of global solutions for a similar reaction-diffusion system with nonlinear boundary condition is proved in [11,12].Further existence and uniqueness of classical solutions of a similar kind of cancer invasion model as (1.1) with taxis effect is studied in [16,30].However, in biological applications, study on lower bound for the finite-time blow-up of solutions is important due to the explosive and diffusive nature of solutions.Further there are some important physical phenomena formulated for biological models with nonlinear boundary conditions rather than the standard Dirichlet boundary conditions.Therefore, in line with these motivations, in this work, we estimate the lower bounds for the finite time blow-up of solutions in R N , N = 2, 3 with Neumann and Robin type boundary conditions for cancer invasion reaction-diffusion system (1.1) using first-order differential inequality techniques.
The paper is organized as follows.In Section 2, we estimate the lower bound for the finitetime blow-up of solutions of (1.1) with suitable auxiliary function in R 3 under Neumann and Robin type boundary conditions.Further, in Section 3, we extend the same results in R 2 by changing certain inequalities.

Lower bounds for finite time blow-up of solutions in R 3
In this section, we consider a parabolic system (1.1) and seek a lower bound on blow-up time for a non-negative solution if it is occur at some finite time t * .In order to obtain the desire result, we first define the suitable auxiliary function for the problem (1.1).Under the assumptions of the Neumann boundary conditions (h i (t) = 0) and Robin boundary conditions (h i (t) > 0) in (1.1), we attain the lower bounds for finite-time blow-up of solutions with help of certain inequalities and the considered auxiliary function.
We define the following auxiliary function to obtain the lower bounds of (u, v, w) for finite time t * : where α(t), β(t) and γ(t) are suitable time dependent positive functions.Further, we define ϕ 0 as Definition 2.1.We say that the triple solution (u, v, w) of (1.1) blows-up in ϕ-measure at time Theorem 2.2 (with Neumann boundary condition).Suppose that (u, v, w) is a non-negative clas- sical solution of (1.1) in a bounded convex domain Ω ⊂ R 3 with origin inside.If the triple solution (u, v, w) becomes unbounded in L 2 (Ω)-norm at t = t * , then t * satisfies the lower bound where Proof.Differentiating (2.1), we have Using zero flux boundary condition and divergence theorem, we get ) Using Hölder's inequality and the standard inequality a r b s ≤ ra + sb, a > 0, b > 0, s + r = 1, we get (2.8) Similarly we get (2.9) Substituting (2.6)-(2.9) in (2.5), we get where Using the Sobolev-type inequality (see Lemma A2 in [18]) and standard inequality (a we can estimate the terms of (2.10) as follows: 1 Ω u 2 dx 3 2 where Similarly we get 1 Ω w 2 dx 3 2 where the constants are defined as before.Substituting (2.11)-(2.12) in (2.10), we get (2.13) Choosing α(t), β(t), γ(t), 0 (t), 1 (t) and 2 (t) as follows: we obtain the following first order differential inequality where If the solution blows up at t * , then there exists a time t 1 ≥ 0 such that ϕ(t) ≥ ϕ 0 , t ≥ t 1 and (  where η 1 (t) is the first eigenvalue of (2.17) and n is the unit normal vector.If the triple solution (u, v, w) becomes unbounded in L 2 (Ω)-norm at t = t * , then t * satisfies the lower bound where R −1 is the inverse function of R(t) := t 0 R(τ)dτ and R := R(α(t), β(t), γ(t), Ω, ϕ 0 ) is a positive function.
Proof.In order to prove the theorem, we use similar arguments as in Theorem 2.2.However, we consider the following inequality for u and w in place of (2.6) and (2.7).
where A 1 (t), A 2 (t) and A 3 (t) are defined as before.From the variational definition of η 1 (t) and (2.17), we get and therefore we have the following inequalities for u and w (2.23) Choosing α(t), β(t), γ(t), 0 (t), 1 (t) and 2 (t) as follows: we obtain the following first order differential inequality where Then similar arguments as in Theorem 2.2 leads to where

Lower bounds for finite time blow-up of solutions in R 2
In this section, we prove a lower bound for the finite-time blow-up of solutions of the cancer invasion parabolic system (1.1) in a bounded domain Ω ⊂ R 2 .As in the previous section, we consider the two cases, the Neumann boundary condition and the Robin type boundary for the parabolic system (1.1) to prove the blow-up of solutions (u, v, w) for some time t * .Theorem 3.1 (with Neumann boundary condition).Suppose that (u, v, w) is a non-negative clas- sical solution of (1.1) in a bounded convex domain Ω ⊂ R 2 with origin inside.Assume further that h i (t) = 0, i = 1, 2, in (1.1).If the triple solution (u, v, w) becomes unbounded in L 2 (Ω)-norm at t = t * , then t * satisfies the lower bound where N −1 is the inverse function of N(t) := t 0 N(τ)dτ and N := N(α(t), β(t), γ(t), Ω, ϕ 0 ) is a positive function.
Proof.The proof of the theorem relies on evaluating the integrals Ω u 3 dx, Ω v 3 dx and Ω w 3 dx in (2.10).We use the following two inequalities (see [18]) in order to achieve our goal.For any f ∈ C 1 (Ω) where Ω is a convex domain in R 2 and ρ 0 , d are defined as before, Using (3.4) and Cauchy's inequality, we get Choosing α(t), β(t), γ(t), 0 (t), 1 (t) and 2 (t) as follows: we obtain the following first order differential inequality where If the solution blows up at t * , then there exists a time t 1 ≥ 0 such that ϕ(t) ≥ ϕ 0 , t ≥ t 1 and (3.9) Replacing (3.9) in (3.8), we get where Theorem 3.2 (with Robin boundary condition).Suppose that (u, v, w) is a non-negative classical solution of (1.1) in a bounded convex domain Ω ⊂ R 2 with origin inside.Further assume that 0 ≤ α (t) < 2α(t)d 1 η 1 (t) and 0 ≤ γ (t) < 2γ(t)d 2 η 1 (t), where η 1 (t) is the first eigenvalue of (2.17).If the triple solution (u, v, w) becomes unbounded in L 2 (Ω)-norm at t = t * , then t * satisfies the lower bound and d = max Ω |x| are positive constants. ) (with Robin boundary condition).Suppose that (u, v, w) is a non-negative classical solution of (1.1) in a bounded convex domain Ω ⊂ R 3 with origin inside.Further assume that 0 t ∈ [t 1 , t * ), * ).