Journal of Evolution Equations

This paper deals with a fully parabolic chemotaxis-growth system with signal-dependent sensitivity $$\left\{\begin{array}{ll}u_t=\Delta u-\nabla\cdot(u\chi(v)\nabla v)+\mu u(1-u), \quad &\quad(x,t)\in\Omega\times (0,\infty),\\ v_{t}=\varepsilon \Delta v+h(u,v), \quad &\quad(x,t)\in \Omega\times (0,\infty),\end{array}\right.$$ under homogeneous Neumann boundary conditions in a bounded domain $${\Omega\subset {\mathbb{R}}^{n} (n\geq1)}$$ with smooth boundary, where $${\varepsilon\in(0,1), \mu>0}$$ , the function $${\chi(v)}$$ is the chemotactic sensitivity and h(u,v) denotes the balance between the production and degradation of the chemical signal which depends explicitly on the living organisms. Firstly, by using an iterative method, we derive global existence and uniform boundedness of solutions for this system. Moreover, by relying on an energy approach, the asymptotic stability of constant equilibria is studied. Finally, we shall give an example to illustrate the theoretical results.

In this paper, we study the behavior in time of the solutions for a class of parabolic problems including the p-Laplacian equation and the heat equation. Either the case of singular or degenerate equations is considered. The initial datum $$u_0$$ is a summable function and a reaction term f is present in the problem. We prove that, despite the lack of regularity of $$u_0$$ , immediate regularization of the solutions appears for data f sufficiently regular and we derive estimates that for zero data f become the known decay estimates for these kinds of problems. Besides, even if f is not regular, we show that it is possible to describe the behavior in time of a suitable class of solutions. Finally, we establish some uniqueness results for the solutions of these evolution problems.

In this paper, we are concerned with the 3-D incompressible micropolar fluid system, which is a non-Newtonian fluid exhibiting micro-rotational effects and micro-rotational inertia. We aim at establishing the global Gevrey analyticity in the critical Besov space. As a first step, inspired by Chemin’s work (J Anal Math 77:27–50, 1999), we construct the global-in-time existence of the strong solutions in a more general Besov space $$\dot{B}^{\frac{3}{p}-1}_{p,q}$$ with $$1\le p<\infty , 1\le q\le \infty $$ . A new effective variable $$R=\nabla \times \omega +\frac{1}{2}\Delta u$$ is introduced at the low frequencies, which allows to eliminate the linear coupling terms $$\nabla \times u$$ and $$\nabla \times \omega $$ , and obtain a global priori estimate. Secondly, observing the parabolic behaviors for u and $$\omega $$ , we would establish Gevrey analyticity based on the work by Bae, Biswas and Tadmor (Arch Ration Mech Anal 205:963–991, 2012) for the incompressible Navier–Stokes equations. The idea of effective velocity is also essential for establishing the Gevrey analyticity. As a by-product, the time-decay estimates on any derivative of solutions are also available for large time.

In this work, we study the existence of solutions to a $$2\times 2$$ non-conservative and non-strictly hyperbolic system in one space dimension related to the dynamics of dislocation densities in crystallography, propagating in two opposite directions. For such systems, existence results are mainly established in the sense of viscosity solutions for Hamilton-Jacobi equations. We study this problem for large initial data using the notions of the theory of conservation laws by constructing entropy solutions through the means of an adapted Godunov scheme, where the associated Riemann problem enjoys new features, more elementary waves than usual, and loss of uniqueness in many cases. The existence is obtained in spaces of functions with bounded fractional total variation $$BV^s$$ , for all $$0