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Global and non-global existence of solutions to a nonlocal and degenerate quasilinear parabolic system

Yujuan Chen (2010)

Czechoslovak Mathematical Journal

The paper deals with positive solutions of a nonlocal and degenerate quasilinear parabolic system not in divergence form u t = v p Δ u + a ∫ Ω u d x , v t = u q Δ v + b ∫ Ω v d x with null Dirichlet boundary conditions. By using the standard approximation method, we first give a series of fine a priori estimates for the solution of the corresponding approximate problem. Then using the diagonal method, we get the local existence and the bounds of the solution ( u , v ) to this problem. Moreover, a necessary and sufficient condition for the non-global existence...

Global existence and polynomial decay for a problem with Balakrishnan-Taylor damping

Abderrahmane Zaraï, Nasser-eddine Tatar (2010)

Archivum Mathematicum

A viscoelastic Kirchhoff equation with Balakrishnan-Taylor damping is considered. Using integral inequalities and multiplier techniques we establish polynomial decay estimates for the energy of the problem. The results obtained in this paper extend previous results by Tatar and Zaraï [25].

Global existence for functional semilinear integro-differential equations

Sotiris K. Ntouyas (1998)

Archivum Mathematicum

In this paper, we study the global existence of solutions for first and second order initial value problems for functional semilinear integrodifferential equations in Banach space, by using the Leray-Schauder Alternative or the Nonlinear Alternative for contractive maps.

Global existence of solutions for the non linear Boltzmann equation of semiconductor physics.

Francisco José Mustieles (1990)

Revista Matemática Iberoamericana

In this paper we give a proof of the existence and uniqueness of smooth solutions for the nonlinear semiconductor Boltzmann equation. The method used allows to obtain global existence in time and uniqueness for dimensions 1 and 2. For dimension 3 we can only assure local existence in time and uniqueness. First, we define a sequence of solutions for a linearized equation and then, we prove the strong convergence of the sequence in a suitable space. The metod relies on the use of interpolation estimates...

Gradient flows of the entropy for jump processes

Matthias Erbar (2014)

Annales de l'I.H.P. Probabilités et statistiques

We introduce a new transport distance between probability measures on ℝ d that is built from a Lévy jump kernel. It is defined via a non-local variant of the Benamou–Brenier formula. We study geometric and topological properties of this distance, in particular we prove existence of geodesics. For translation invariant jump kernels we identify the semigroup generated by the associated non-local operator as the gradient flow of the relative entropy w.r.t. the new distance and show that the entropy is...

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