We consider the Cauchy problem for a doubly nonlinear parabolic equation with inhomogeneous source. The existence and uniqueness of solutions are obtained when the initial datum is merely a function in $L_{loc}^1(R^N)$ or a Radon measure.

We investigate the existence of nonnegative solutions for a nonlinear problem involving the fractional $p$-Laplacian operator. The problem is set on a unbounded domain and compactness issues have to be handled.

The basic model for incompressible two-phase flows with phase transitions where the interface is nearly flat in the case of non-equal densities is considered. The local well-posedness of the model in $L_p$-setting was proved in Prüss and Shimizu [14]. In this paper, we prove local well-posedness of the model $L_p$ in time $L_q$ in space settting.

Here, we consider a generalized heat equation $\partial_t \rho=\frac{d}{dx}\frac{d}{dW}\rho$, where $W$ is a finite measure on the one dimensional torus, and $\frac{d}{dW}$ is the Radon-Nikodym derivative with respect to $W$. Such an equation has appeared in different contexts, one of which being related to physical systems and representing a large class of classical and non-classical parabolic equations. As a natural assumption on $W$, we require that the Lebesgue measure is absolutely continuous with respect to $W$. The main result here presented consists of proving, for a suitable topology, a continuous dependence of the solution $\rho$ as a function of $W$.

In this article, we study the well-posedness (uniqueness and existence of solutions) of nonlinear elliptic Partial Differential Equations (PDEs) on a finite graph. These results are obtained using the discrete comparison principle and connectivity properties of the graph. This work is in the spirit of the theory of viscosity solutions for partial differential equations. The equations include the graph Laplacian, the $p$-Laplacian, the Infinity Laplacian, and the Eikonal operator on the graph.

We prove that the critical problem for the fractional Laplacian in an annular type domain, admits a positive solution provided that the inner hole is sufficiently small.

By using variational methods, we study the following doubly critical elliptic system: \[ \begin{cases} -\Delta u-\mu_1\frac{u}{|x|^2}-|u|^{2^{*}-2}u= h(x)\alpha|u|^{\alpha-2}|v|^\beta u\quad & \rm{in}\; \mathbb R^N,\\ -\Delta v-\mu_2\frac{u}{|x|^2}-|v|^{2^{*}-2}v= h(x)\beta |u|^{\alpha}|v|^{\beta-2}v\quad & \rm{in}\; \mathbb R^N, \\ \end{cases} \] where $\alpha >1, \beta >1$ satisfying $\alpha+\beta \leq 2^{*}:=\frac{2N}{N-2}\; (N\geq 3)$, and either $\mu_1=\mu_2=0$ or $\mu_1,\mu_2\in (0,\frac{(N-2)^2}{4})$. The function $h(x)$ is allowed to be sign-changing and satisfies weaker conditions which permit the nonlinearities to include a large class of indefinite weights. We obtain the existence of the ground state solutions. The indefiniteness of $h$ makes the problem intrinsically complicated. However, our assumptions on $h$ are almost optimal.

In this article we consider the third-grade fluids equation in one space dimension. We show that the large time behavior of solutions is given by the very singular self-similar solutions of the associated Burgers equation. The proof of our result relies on the use of fine energy estimates in weighted spaces rewritten in the variables $\xi/\sqrt{\tau}$ and $\ln \tau$.