The exponential decay of weak solutions towards the constant steady state of general cross-diffusion systems in bounded domains with no-flux boundary conditions is investigated. The proof of quantitative decay rates is based on the relative entropy method and involves two parameters: the exponent determining the entropy density and the exponent in the entropy production integral. The key ingredient are novel convex Sobolev inequalities, which extend existing results in the literature to a broad range of values of the two parameters. These inequalities are established using convexity arguments and the Gagliardo-Nirenberg inequality.
We study evolutions of measures under flows of solutions to stochastic differential equations with interaction. Under suitable regularity
assumptions on the coefficients, we prove moment bounds for the compact-set Jacobian norm. These bounds yield pathwise transfer of logarithmic
Sobolev, concentration, and Talagrand...
Kyrylo Kuchynskyi· Theory of Stochastic Process...· 0 citations
For diffusions whose coefficients are smooth, with at most linear growth and bounded derivatives of higher order, we establish explicit pointwise bounds on the spatial derivatives of the classical solution to the Cauchy problem with polynomially growing initial data. We also obtain explicit growth estimates for the der...
We study the fundamental solution operators associated to Kolmogorov equations with symmetric non-local diffusion with rough kernels. We prove that these operators are induced by measurable kernels and prove sharp upper bounds in terms of a kinetic profile governed by the interplay of transport and non-local diffusion....
We consider the Cauchy problem driven by nonlinear fractional mobile-immobile equations (Fr-MIMEs) involving time-varying delays in Hilbert spaces. Based on fixed-point arguments, smoothness of the resolvent operators, and local estimates, we show global solvability and regularity results for both linear and semilinear...
Tuan Van Tran, D. Nguyen· Rocky Mountain Journal of Ma...· 0 citations
We consider the two-species Lotka-Volterra competition-diffusion system where one species has a small diffusion rate relative to the other species and has a small competition coefficient. By the geometric singular perturbation theory, we prove the existence of the wavefront connecting the coexistence state to the trivi...
Chueh-Hsin Chang, Tzi-Sheng Yang· 0 citations
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