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We analyze the mathematical properties of a multi-species biofilm cross-diffusion model together with very general reaction terms and mixed Dirichlet-Neumann boundary conditions on a bounded domain. This model belongs to the class of…
We provide a unified viewpoint on two illposedness mechanisms for dispersive equations in one spatial dimension, namely degenerate dispersion and (the failure of) the Takeuchi--Mizohata condition. Our approach is based on a robust energy-…
In this paper, we consider the well-posedness of the Cauchy problem for a physical model of the extrusion process, which is described by two systems of conservation laws with a free boundary. By suitable change of coordinates and fixed…
In the present paper, we consider the Cauchy problem of a system of quadratic derivative nonlinear Schr\"odinger equations which was introduced by M. Colin and T. Colin (2004) as a model of laser-plasma interaction. The local existence of…
In this paper, we consider the global well-posedness and time-decay rates of solution to the Cauchy problem for 3D convective Cahn-Hilliard equation with double-well potential via a refined pure energy method. In particular, the optimal…
In this paper, we establish the well-posedness of Cauchy problems for weak solutions to second-order degenerate parabolic equations with a non-smooth, time-dependent degenerate elliptic part that includes both bounded and unbounded…
The short pulse equation provides a model for the propagation of ultra-short light pulses in silica optical fibers. It is a nonlinear evolution equation. In this paper the wellposedness of bounded solutions for the homogeneous initial…
This paper is concerned with the strong solution to the Cauchy-Dirichlet problem for backward stochastic partial differential equations of parabolic type. Existence and uniqueness theorems are obtained, due to an application of the…
In this paper we establish optimal solvability results, that is, maximal regularity theorems, for the Cauchy problem for linear parabolic differential equations of arbitrary order acting on sections of tensor bundles over boundaryless…
We consider the Cauchy problem for a class of nonlinear degenerate parabolic equa- tion with forcing. By using the vanishing viscosity method we obtain generalized solutions. We prove some regularity results about this generalized…
We study the Cauchy problem for one-dimensional dispersive equations posed on $\mathbb{R} $, under the hypotheses that the dispersive operator behaves, for high frequencies, as a Fourier multiplier by $ i |\xi|^\alpha \xi $ with $ 1 \le…
This paper is concerned with reaction-diffusion-advection equations in spatially periodic media. Under an assumption of weak stability of the constant states 0 and 1, and of existence of pulsating traveling fronts connecting them, we show…
We present a general method of solving the Cauchy problem for multidimensional parabolic (diffusion type) equation with variable coefficients which depend on spatial variable but do not change over time. We assume the existence of the…
We study the long time behavior of solutions of the Cauchy problem for nonlinear reaction-diffusion equations in one space dimension with the nonlinearity of bistable, ignition or monostable type. We prove a one-to-one relation between the…
We consider the Cauchy problem for doubly non-linear degenerate parabolic equations on Riemannian manifolds of infinite volume, or in $\R^N$. The equation contains a weight function as a capacitary coefficient which we assume to decay at…
We study the Cauchy problem for nonlocal reaction diffusion equations with bistable nonlinearity in 1D spatial domain and investigate the asymptotic behaviors of solutions with a one-parameter family of monotonically increasing and…
We study the strong solvability of the Cauchy-Dirichlet problem for parabolic quasilinear equations with discontinuous data. The principal coefficients depend on the point $(x,t)$ and on the solution u, the dependence on x is of VMO type…
We study the existence and qualitative properties of solutions to the Cauchy problem associated to the quasilinear reaction-diffusion equation $$ \partial_tu=\Delta u^m+(1+|x|)^{\sigma}u^p, $$ posed for $(x,t)\in\real^N\times(0,\infty)$,…
In this paper, we derive the large-time profile of solutions to the Cauchy problem of a hyperbolic-parabolic system modeling the vasculogenesis in $\R^3$. When the initial data are prescribed in the vicinity of a constant ground state, by…
This paper investigates the nonlinear Schr\"{o}dinger equation with a singular convolution potential. It demonstrates the local well-posedness of this equation in a modified Sobolev space linked to the energy. Additionally, we derive…