Abstract
We consider nonlinear evolution equations with logistic term satisfying initial Neumann-boundary condition and show global existence in time of solutions to the problem in arbitrary space dimension by using the method of energy. Applying the result to a mathematical model of tumour invasion, we discuss the property of the rigorous solution to the model. Finally we will show the time depending relationship and interaction between tumour cells, the surrounding tissue and matrix degradation enzymes in the model by computer simulations. It is seen that our mathematical result of the existence and asymptotic behaviour of solutions verifies our simulations, which also confirm the mathematical result visibly.
Highlights
In this paper we consider the initial Neumann-boundary value problem of nonlinear evolution equations with logistic term, arising from tumour invasion models with proliferation and re-establishment: (NE)( ( ) ) ∂t2u= D∆ut + ∇ ⋅ χ ut, e−u e−u∇u + μ (1− ut )ut in Ω × (0,T ) (1)= ∂∂ν u ∂Ω 0 on ∂Ω × (0,T ) (2)= u ( x, 0) u= 0 ( x), ut ( x, 0) u1 ( x) in Ω (3) ( ) ∂ ∂∂t = ∂t, ∂xi = ∂xi, i = 1, ⋅ ⋅ ⋅, n, ∇u = ∂x1u, ∂xnu ∇2u = ∇ ⋅ ∇u ∆u
We study the case of μ1, μ2 > 0, which describes tumour invasion phenomena with tumour cell proliferation and re-establishment of extracellular matrix (ECM) respectively
In this paper first we show the rigorous mathematical result of (C-L) and computer simulations, of which the validity is guaranteed by our mathematical result
Summary
In this paper we consider the initial Neumann-boundary value problem of nonlinear evolution equations with logistic term, arising from tumour invasion models with proliferation and re-establishment: (NE). 2 xn u where u := u ( x, t ) for ( x, t ) ∈ Ω × (0,T ) , D and μ are positive constants, Ω is a bounded domain in Rn and ∂Ω is a smooth boundary of Ω and ν is the outer unit normal vector. The eigenvalues of −∆ with the homogeneous Neumann boundary conditions are denoted by {λi | i = 0,1, 2, } satisfying =0 λ0 < λ1 ≤ → +∞ and φi = φi ( x) indicates the L2 normalized eigenfunction corresponding to λi. Putting u ( x, t ) =a + bt + v ( x, t ) in (1) it follows from (1) that ( ( ) ) = ∂t2v D∆vt + ∇ ⋅ χ b + vt , e−a−bt−v e−a−bt−v∇v + μvt (t ) (1 − 2b − vt ) + μb (1 − b) where a and b are positive parameters. Where n := n ( x, t ) is the density of tumour cells, m := m ( x, t ) is matrix degradation enzymes (MDEs) concentration and f := f ( x, t ) is extracellular matrix (ECM) density in Ω × (0,T ) and dn , γ , μ1 , η , μ2 , dm , α and β are positive constants
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