This paper is concerned with the initial-boundary value problem to 3D double-diffusive convection system. First, we establish the global weak solutions for the double-diffusive convection system in a domain $\Omega$ with Navier boundary condition by Galerkin approximation method. Then by making higher-order estimates of the approximation function with higher regularity assumption on the initial data, combined with compactness argument we obtain the local strong solution with uniqueness. Secondly, we obtain the classical Serrin-type blow-up criterion for the local strong solution in the Lorentz space. Lastly, we focus on global stability of strong large solutions for the double- diffusive convection system with Navier boundary conditions, it is shown that the large solution is stable with the additional assumption on some suitable integrable property of solution.

In this paper, explicit solutions of a class of generalized space-time fractional Cauchy problems with time-variable coefficients are given. The representation of a solution involves kernels given by convergent infinite series of fractional integro-differential operators, which can be extensively and efficiently applied for analytic and computational goals. Time-fractional operators of complex orders with respect to a given function are used. Further, we study inverse Cauchy problems of finding time dependent coefficients for fractional wave and heat type equations, which involve the explicit representation of the solution of the direct Cauchy problem and a recent method to recover variable coefficients for the considered inverse problems. Concrete examples and particular cases of the obtained results are discussed.

This paper is devoted to study a class of $p(x)$-Laplacian equation involving concave-convex nonlinearities. By means of a perturbation technique, using variational method and a priori estimation, the existence of two positive solutions to this equation is obtained.

The aim of this paper is to study the following fractional Laplacian problems with logarithmic and critical nonlinearities: \begin{align*} \begin{cases} (-\Delta)^{s}u=\lambda a(x)u\ln|u|+\mu|u|^{2_s^*-2}u \ \ & x\in\Omega,\\ u=0\ \ & x\in\mathbb{R}^N\setminus\Omega, \end{cases} \end{align*} where $s\in(0,1),\ N > 2s,$ $\lambda,\mu > 0$, $\Omega\subset \mathbb{R}^N$ is a bounded domain with Lipschitz boundary, $a\in L^\infty(\Omega)$ is a sign-changing function, $2_s^*$ is the critical Sobolev exponent and $(-\Delta)^{s}$ is the fractional Laplacian. When $\lambda$ and $\mu$ satisfy suitable assumptions, the existence and multiplicity of nontrivial and nonnegative continuous solutions for the above problem are obtained by applying the Nehari manifold approach.