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We study a system of partial differential equations describing a steady thermoconvective flow of a non-Newtonian fluid. We assume that the stress tensor and the heat flux depend on temperature and satisfy the conditions of p,q-coercivity with $p>\frac{2n}{n+2}$, $q>\frac{np}{p(n+1)-n}$, respectively. Considering Dirichlet boundary conditions for the velocity and a mixed and nonlinear boundary condition for the temperature, we prove the existence of weak solutions. We also analyze the existence and uniqueness of strong solutions for small and suitably regular data.

This article concerns the Cauchy problem associated with the nonlinear fourth-order Schrodinger equation with isotropic and anisotropic mixed dispersion. This model is given by the equation $$ i\partial_tu+\epsilon \Delta u+\delta A u+\lambda|u|^\alpha u=0,\quad x\in\mathbb{R}^{n},\; t\in \mathbb{R}, $$ where A is either the operator $\Delta^2$ (isotropic dispersion) or $\sum_{i=1}^d\partial_{x_ix_ix_ix_i}$, $1\leq d

This paper is devoted to the theoretical analysis of a coupled nonlinear system of fractional Schrödinger equations in Rn×R+, n≥1, considering time fractional derivative in the Caputo sense, and a fractional spatial dispersion defined in terms of the Fourier transform. We prove the existence of local and global mild solutions, as well as the asymptotic stability of global mild solutions, considering power-type nonlinearities and initial data in the framework of weak-Lp spaces, which contain singular functions with infinite energy. As consequence of the embedding of weak-Lp spaces in Lloc2, for p>2, the obtained solutions have finite local L2-mass. In addition, we discuss the scenario in which it is possible to obtain the existence of self-similar solutions, which is a symmetric property that reproduces the structure of physical phenomena in different spatio-temporal scales. Our results are applicable, in the fractional setting, to the nonlinear Schrödinger and Biharmonic equations, as well as in a large class of dispersive systems appearing in nonlinear optics.

We derive new results about existence and uniqueness of local and global solutions for the nonlinear Schrödinger equation, including self-similar solutions. Our analysis is performed in the framework of weak-$L^{p}$ spaces. [ABSTRACT FROM AUTHOR]