This paper deals with the Cauchy problem for a doubly degenerate parabolic equation with inhomogeneous source and initial data measures. The existence of a solution is established under the optimal assumption on initial data measure.

In this paper we show the existence of $\varphi$ satisfying the estimate $$ \|\varphi-\operatorname{id}\|_{C^{r+1,\alpha}} \leq C\|g\|_{C^{r+1,\alpha}}\|f-g\|_{C^{r,\alpha}} $$ together with the Jacobian equation $$ g\circ \varphi \det\nabla \varphi=f. $$ We, moreover, prove that the previous estimate is sharp.

In this paper, we adapt and extend B. Birnir, G. Ponce, and N. Svanstedt result related with local ill-posedness for the Cauchy problem to the mKdV equation in [5] to the generalized $p$-Gardner equation. We prove that the Cauchy problem for the generalized $2$-Gardner equation is ill-posed in the Sobolev space $H^s(\mathbb{R})$ for $s < -\frac12$.

We prove that the $L^{\infty}$ end-point Kato-Ponce inequality (Leibniz rule) holds for the fractional Laplacian operators $D^s=(-\Delta)^{s/2}$, $J^s=(1-\Delta)^{s/2}$, $s>0$. This settles a conjecture by Grafakos, Maldonado and Naibo [7]. We also establish a family of new refined Kato-Ponce commutator estimates. Some of these inequalities are in borderline spaces.

A nonlocal diffusion population model with spatio-temporal delays is considered in this paper. It is shown that this nonlocal diffusion equation admits traveling wave solutions. The uniqueness and minimal wave speed of the traveling wave solutions are obtained. Furthermore, the effects of different dispersal strategies on the minimal wave speed are characterized.

We consider nonlinear periodic problems driven by the sum of a scalar $p$-Laplacian and a scalar Laplacian and a Carath\'{e}odory reaction, which at $\pm\infty$, is resonant with respect to any higher eigenvalue. Using variational methods, coupled with suitable perturbation and truncation techniques and Morse theory, we prove a three solutions theorem. For equations resonant with respect to the principal eigenvalue $\hat \lambda_0=0$, we establish the existence of nodal solutions.

In this paper, we study the class of non-linear viscoelastic equation with lower order perturbation of $\overrightarrow{p}(x,t)-$ Laplacian type and a memory term \[ u_{tt}+\Delta^{2}u-\Delta_{\overrightarrow{p}(x,t)}u+\int_{0}^{t}g(t-s)\Delta u(s)ds -\epsilon\Delta u_{t}+f(u) =0\text{, } \] \[ (x,t)\in Q_{T}=\Omega\times(0,T),\quad\Omega\in\mathbb{R}^{n}. \] We prove local and global existence and uniqueness of weak solutions. These results are obtained assuming a strong damping $\epsilon\Delta u_{t}$ $(\epsilon>0)$ acting in the domain and provided the memory term decays exponentially and $f(u)$ is a nonlinear perturbation.

In this paper, we obtain a multiplicity result for a nonhomogeneous elliptic problem involving critical growth and an improved Hardy potential. The Moser inequality and Hardy inequality due to Adimurthi and Sandeep, are combined with critical point

The mathematical analysis of two one-phase unidimensional and non-classical Stefan problems with nonlinear thermal coefficients is obtained. Two related cases are considered, one of them has a temperature condition on the fixed face $x=0$ and the other one has a flux condition of the type $-q_{0}/\sqrt{t }$ $ ( q_{0}>0 ) .$ In the first case, the source function depends on the heat flux at the fixed face $x=0,$ and in the other case it depends on the temperature at the fixed face $x=0. $ In both cases, we obtain sufficient conditions in order to have the existence of an explicit solution of a similarity type, which is given by using a double fixed point.