Narrow Results

In this paper, we consider the nonlinear cauchy problem

In this paper, we consider solving a system of fractional differential equations with nonlinear source term. The nonlocal source term is a convolution of an exponential with a kernel. The solved system is able to be seen as a generalization of a number of varied systems of equations with solutions that blows up in a finite time. Under initial conditions, the nontrivial global solutions blow-up and certain assumptions are established by the weak formulation of the problem with some estimation inequalities.

This paper proposes a review focused on exotic chemotaxis and cross-diffusion models in complex environments. The term exotic is used to denote the dynamics of models interacting with a time-evolving external system and, specifically, models derived with the aim of describing the dynamics of living systems. The presentation first, considers the derivation of phenomenological models of chemotaxis and cross-diffusion models with particular attention on nonlinear characteristics. Then, a variety of exotic models is presented with some hints toward the derivation of new models, by accounting for a critical analysis looking ahead to perspectives. The second part of the paper is devoted to a survey of analytical problems concerning the application of models to the study of real world dynamics. Finally, the focus shifts to research perspectives within the framework of a multiscale vision, where different paths are examined to move from the dynamics at the microscopic scale to collective behaviors at the macroscopic scale.

A novel class of implicit constitutive relations is studied, wherein the stress and the linearized strain appear linearly, that describe material response in elastic porous bodies like rocks, ceramics, concrete, cement, bones and metals. The constitutive relation is applied to a body with a crack subjected to non-penetration conditions between the opposite crack faces. To treat well-posedness of a corresponding variational inequality, we rely on a new approximation by thresholding dilatation and apply the Lions existence theorem on pseudo-monotone variational inequalities. An analytical solution to a specific example (without crack) under uniform triaxial loading is constructed, wherein blow-up can take place at a finite load, and this difficulty is overcome within a thresholding approximation so that blow-up does not occur.

This paper is devoted to the study of the behaviors of the solution to Fujita-type equations on finite graphs. Under certain conditions given by absorption term of the equations, we prove respectively local existence and blow-up results of solutions to Fujita-type equations on finite graphs. Our results contain some previous results as special cases. Finally, we provide some numerical experiments to illustrate the applicability of the obtained results.

In this paper, we study the blow-up problem for Fujita-type equations with a general absorption term $f$ on finite graphs and locally finite graphs, respectively. We prove that if $f$ satisfies appropriate conditions, then the solutions of the equation blow up in finite time. The obtained results are generalization of those given by the author in previous papers. At the end of the paper, we provide some numerical experiments to illustrate the applicability of the obtained results.

In this paper, we deal with the blow-up problems for Fujita-type parabolic system involving time-dependent coefficients on graphs. Under appropriate conditions, we prove that the nonnegative solution of the parabolic system blows up in a finite time on finite graphs and locally finite graphs, respectively. The results obtained extend some previous results of [Y. Lin and Y. Wu, Blow-up problems for nonlinear parabolic equations on locally finite graphs, Acta Math. Sci. Ser. B 38(3) (2018) 843–856; Y. Wu, Local existence and blow-up of solutions to Fujita-type equations involving general absorption term on finite graphs, Fractals 30(2) (2022) 2240053].

In this paper, we present a blow-up result for large data for relaxed compressible Navier–Stokes models avoiding the possibility of reaching the boundary of hyperbolicity. Thus, a previous result is improved and further examples are given illustrating possible effects of a relaxation and contrasting the classical compressible Navier–Stokes equations without relaxation where solutions for large data exist globally.

In this paper, we study the periodic Hunter–Saxton equation with weak dissipation. We first establish the local existence of strong solutions, blow-up scenario and blow-up criteria of the equation. Then, we investigate the blow-up rate for the blowing-up solutions to the equation. Finally, we prove that the equation has global solutions.

We consider a periodic 2-component Camassa–Holm equation with vorticity in the paper. We first give the local well-posedness and the blow-up criterion for strong solutions to the equation in the Sobolev space H^s, . We then present a global existence result for strong solutions to the equation. We finally obtain several blow-up results and the blow-up rate of strong solutions to the equation. We finally examine the propagation behaviour of compactly supported solutions to the equation. The obtained results cover and improve the earlier results for a periodic 2-component Camssa–Holm equation without vorticity.

In this paper, we consider a degenerate hyperbolic inequality in an exterior domain under three types of boundary conditions: Dirichlet-type, Neumann-type, and Robin-type boundary conditions. Using a unified approach, we show that all the considered problems have the same Fujita critical exponent. Moreover, we answer some open questions from the literature regarding the critical case.