Existence of T-solution for degenerated problem via Minty’s lemma

We study the existence result of solutions for the nonlinear degenerated elliptic problem of the form, -div（a（x, u,△↓u）） = F in Ω, where Ω is a bounded domain of R^N, N≥2, a ：Ω×R×R^N→R^N is a Carathéodory function satisfying the natural growth condition and the coercivity condition, but they verify only the large monotonicity. The second term F belongs to W^-1,p′（Ω, w^＊）. The existence result is proved by using the L^1-version of Minty＇s lemma.     

Entropy solutions for nonlinear nonhomogeneous Neumann problems involving the generalized p(x)-Laplace operator

In this work we investigate a class ofnonlinear p(x) Laplace problems with Neumann nonhomoge-neous boundary conditions and L1 data. The techniques of entropy solutions for elliptic equationsare used to prove the existence of a solution.     

Renormalized solutions to nonlinear parabolic problems with blowing up coefficients and general measure data

An existence result is established for a class of quasilinear parabolic problem which is a diffusion type equations having continuous coefficients blowing up for a finite value of the unknown, a second hand \(\mu \in \mathcal {M}_{b}(Q)\) and an initial data \(u_{0}\in L^{1}(\Omega )\). We develop a technique which relies on the notion of a renormalized solution and an adequate regularization in time for certain truncation functions. Some compactness results are also shown under additional hypotheses.

     

On nonlinear elliptic equations with Hardy potential and L^{1}-data

We consider a class of nonlinear elliptic equations involving the Hardy potential and lower order terms whose simplest model is $$\begin{aligned} -\Delta u +b(|u|)|\nabla u|^{2}+\nu |u|^{s-1}u=\lambda \frac{u}{|x|^{2}}+f \end{aligned}$$ in a bounded open $\varOmega $ of $\mathbf{R }^{N}, N\ge 3,$ containing the origin, $s>\frac{N}{N-2}, \nu $ and $\lambda $ are positive real numbers. We prove that the presence of the term $\nu |u|^{s-1}u$ has an effect on the existence of solutions when $f\in L^{1}(\varOmega )$ assuming only that $b\in L^{1}(\mathbf{R })$ without any sign condition (i.e. $b(s)s\ge 0$ ).     

Generalized p(x)-elliptic system with nonlinear physical data

This paper considers the following Dirichlet problem of the form \[-\text{div}\,\big(\Phi(Du-\Theta(u)\big)=v(x)+f(x,u)+\text{div}\,\big(g(x,u)\big),\] which corresponds to a diffusion problem with a source $v$ in moving and dissolving substance, the motion is described by $g$ and the dissolution by $f$. By the theory of Young measure we will prove the existence result in variable exponent Sobolev spaces $W^{1,p(x)}_0(\Omega;\R^m)$.     

Exploring the interplay between memory effects and vesicle dynamics: A five-dimensional analysis using rigid sphere models and mapping techniques

The memory effect in vesicle dynamics refers to the persistence of shape changes in lipid vesicles, a type of lipid bilayer membrane that mimics some features of real cells, particularly red blood cells (RBCs). To study this effect, a fractional rigid sphere model in five dimensions has been investigated, which maps the dynamics of a vesicle using the Caputo operator. This model provides new insights into the mechanisms behind the memory effect as well as the emergence of two other motions, namely, tank-treading with underdamped and tank-treading with overdamped vesicle oscillations, in addition to the tank-treading mode where the vesicle rotates about its axis while maintaining a constant shape. Overall, this work represents a significant advance in our understanding of vesicle dynamics and has important implications for understanding the behavior of RBCs.     

Quasilinear elliptic systems with nonstandard growth and weak monotonicity

We prove the existence of solutions for a quasilinear elliptic system $$\begin{aligned} \left\{ \begin{array}{ll} -\text {div}\,\sigma (x,u,Du)&{}=f(x,u,Du)\quad \text {in}\;\varOmega ,\\ u&{}=0\quad \text {on}\;\partial \varOmega . \end{array} \right. \end{aligned}$$The results are obtained in Orlicz–Sobolev spaces by means of the Young measures.