We determine the shape which minimizes, among domains with given measure, the first eigenvalue of the anisotropic laplacian perturbed by an integral of the unknown function. Using also some properties related to the associated \lq\lq twisted\rq\rq problem, we show that, this problem displays a saturation phenomenon: the first eigenvalue increases with the weight up to a critical value and then remains constant.

We prove a Liouville type result for bounded, entire solutions to a class of variational semilinear elliptic systems, based on the growth of their potential energy over balls with growing radius. Important special cases to which our result applies are the Ginzburg-Landau system and systems that arise in the study of multi-phase transitions.

Results on the local and global boundedness of nonnegative weak subsolutions of the doubly nonlinear parabolic equation $$ (u^{q})_t-\text{div}\,{(|\nabla u|^{p-2}\nabla u)}=0, $$ are obtained for $p > 1$ and $0 < q < 1$, that is, for equations presenting a singularity in the time derivative part (as well as a singularity, $1 < p < 2$, or degeneracy, $p > 2$, in the principal part of the operator). We work in measure spaces equipped with a doubling non-trivial Borel measure supporting a Poincaré inequality.

This paper concerns linear first-order hyperbolic systems in one space dimension of the type $$ \partial_tu_j + a_j(x,t)\partial_xu_j + \sum\limits_{k=1}^nb_{jk}(x,t)u_k = f_j(x,t),\; x \in (0,1),\; j=1,\ldots,n, $$ with periodicity conditions in time and reflection boundary conditions in space. We state a non-resonance condition (depending on the coefficients $a_j$ and $b_{jj}$ and the boundary reflection coefficients), which implies Fredholm solvability of the problem in the space of continuous functions. Further, we state one more non-resonance condition (depending also on $\partial_ta_j$), which implies $C^1$-solution regularity. Moreover, we give examples showing that both non-resonance conditions cannot be dropped, in general. Those conditions are robust under small perturbations of the problem data. Our results work for many non-strictly hyperbolic systems, but they are new even in the case of strict hyperbolicity.

In this article, we study a coupled Allen-Cahn-Navier-Stokes model with delays in a two-dimensional domain. The model consists of the Navier-Stokes equations for the velocity, coupled with a Allen-Cahn model for the order (phase) parameter. We prove the existence of an attractor using the theory of pullback attractors.

In this note we consider local invariant manifolds of functional differential equations $x^{\prime}(t)=f(x_{t})$ representing differential equations with state-dependent delay. Starting with a local center-stable and a local center-unstable manifold of the functional differential equation at a stationary point, we construct, by a straightforward application of the Implicit Mapping Theorem, a local center manifold.

In this paper, we consider some integral systems in the half space $\mathbb R^N_+$ and obtain Liouville type theorems about the positive solutions. By moving plane method in terms of the integral form, we shall see that the positive solution $(u(x_1,...,x_N), v(x_1,...,x_N))$ of the integral systems must be independent of the first $(N-1)$-variables, i.e., $u=u(x_N),v=v(x_N)$. Then, combine with the order estimates about $x_N$, we reduce the problem to a sequence of algebraic systems. Furthermore, we discuss the relationship between the integral system and the fractional differential system related to the fractional Lane-Emden equations. By this way, we obtain two non-existence theorems for the fractional differential system.

In this paper, we discuss the Cauchy problem for a degenerate parabolic-hyperbolic equation with a multiplicative noise. We focus on the existence of a solution. Using nondegenerate smooth approximations, Debussche, Hofmanová and Vovelle [8] proved the existence of a kinetic solution. On the other hand, we propose to construct a sequence of approximations by applying a time splitting method and prove that this converges strongly in $L^1$ to a kinetic solution. This method will somewhat give us not only a simpler and more direct argument but an improvement over the existence result.