Abstract
We prove the existence of solutions of degenerate parabolic-parabolic Keller-Segel system with no-flux and Neumann boundary conditions for each variable respectively, under the assumption that the total mass of the first variable is below a certain constant. The proof relies on the interpretation of the system as a gradient flow in the product space of the Wasserstein space and the standard \begin{document}$L^2$\end{document} -space. More precisely, we apply the ''minimizing movement'' scheme and show a certain critical mass appears in the application of this scheme to our problem.
Highlights
We consider the following degenerate parabolic system: ∂tu = ∇ · (∇um − u∇v), x ∈ Ω, t > 0,∂tv = ∆v − γv + u, x ∈ Ω, t > 0, (1)
We focus on m = 2 − 2/d and impose the following boundary conditions:
In parabolic-parabolic system, Blanchet and Laurencot [8] prove the timeglobal existence of solutions under the assumption that M < Mc, where Mc is the same quantity as the threshold mass in parabolic-elliptic case mentioned above
Summary
We consider the following degenerate parabolic system:. u(x, 0) = u0(x), v(x, 0) = v0(x), x ∈ Ω, where γ is a non-negative constant and Ω is a bounded domain in Rd, d > 2, with smooth boundary. We consider the following degenerate parabolic system:. We focus on m = 2 − 2/d and impose the following boundary conditions:. Notice that (1) preserves the mass of u over Ω by the boundary conditions (2). We are interested in non-negative solutions to (1). One of the fundamental questions concerning the system (1) is whether or not a threshold mass that separates the existence from the non-existence of blow-up solutions exists. We are interested in the existence of the following threshold mass Mc: if M < Mc, all solutions exist globally in time, while for any M > Mc there exists a solution with u0 L1 = M that blows up in finite time.
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