Browsing by Author "Rani, Poonam"

We consider the following chemotaxis-haptotaxis system: {ut=∇⋅(D(u)∇u)−χ∇⋅(S(u)∇v)−ξ∇⋅(u∇w),x∈Ω, t>0,vt=Δv−v+u,x∈Ω, t>0,wt=−vw,x∈Ω, t>0, under homogeneous Neumann boundary conditions in a bounded domain Ω⊂Rⁿ,n≥3 with smooth boundary. It is proved that for [Formula presented]≤A(s+1)^α for α<[Formula presented] and under suitable growth conditions on D, there exists a uniform-in-time bounded classical solution. Also, we prove that for radial domains, when the opposite inequality holds, the corresponding solutions blow-up in finite or infinite-time. We also provide the global-in-time existence and boundedness of solutions to the above system with small initial data when D(s)=1,S(s)=s.

In this article, we consider the following parabolic-parabolic-ODE minimal chemotaxis-haptotaxis system {ut=Δu−χ∇⋅(u∇v)−ξ∇⋅(u∇w),x∈Ω, t>0,vt=Δv−v+u,x∈Ω, t>0,wt=−vw,x∈Ω, t>0,[Formula presented]=0,x∈∂Ω, t>0,u(x,0)=u0(x),v(x,0)=v0(x),w(x,0)=w0(x),x∈Ω, in a bounded domain Ω⊂Rⁿ,n⩾3 with smooth boundary. We show that the finite time blow-up occurs to the above system. More specifically, we establish that in a radial setting, the generic mass blow-up phenomenon observed in corresponding chemotaxis-only systems (obtained by setting w≡0) is preserved in chemotaxis-haptotaxis system. Our proof demonstrates that for a given initial mass, there exists radially symmetric positive initial data for which the corresponding solution blows-up. Moreover, we illustrate that such initial data constitute a considerably large set in the sense that it is dense in C⁰(Ω‾)×W^1,∞(Ω)×C²(Ω‾) with respect to topology in L^p(Ω)×W^1,2(Ω)×L^∞(Ω), with p∈(1,[Formula presented]). Analogous results for corresponding parabolic-elliptic-ODE system [53] are already available, where the parabolic-parabolic-ODE case was open for further study.

We consider the following chemotaxis system: (Formula presented.) under homogeneous Neumann boundary conditions in a bounded smooth domain Ω⊂Rn,n=2,3 with nonlinear functions f and g. We establish the existence of a global classical solution under the smallness assumption on initial data. This result generalizes the existing findings for the minimal case, where f(s)=s and g(s)=s. We further present blow-up criteria for the system.

We consider the following Keller-Segel system with gradient dependent chemotactic coefficient: {ut = ∆u − χ∇ · (uf(|∇v|)∇v), 0 = ∆v − v + g(u), in smooth bounded domains Ω ⊂ Rⁿ, n ≥ 1 with f(ξ) = (ξ^p−2(1+ξ^p)^{q− p/p}), 1 < q ≤ p < ∞ and g(ξ) = ξ/(1+ξ)^1-β, ξ ≥ 0, β ∈ [0, 1]. We show the existence of a global weak solution, bounded in L^∞-norm, if 1 < q ≤ p {< ∞, n = 1, 1 < q < n−ⁿ1 , n ≥ 2.

We consider fully parabolic two-species chemotaxis system with (Formula presented.) -Laplacian diffusion in a smooth bounded domain (Formula presented.) with (Formula presented.) We show the existence of globally bounded weak solutions under the assumption that (Formula presented.) -norm of (Formula presented.) is bounded by a universal constant. We first get time-independent bounds for solution components of the approximate system. Then, pass the limit using Aubin–Lions lemma to get the solution candidate.