The global existence theorem of classical solutions for one-dimensional nonlinear thermoelasticity is proved for small and smooth initial data in the case of a bounded reference configuration for a homogeneous medium, considering the Neumann type boundary conditions: traction free and insulated. Moreover, the asymptotic behaviour of solutions is investigated.

We consider a generic scalar model for the Oseen equations in an exterior three-dimensional domain. We assume the case of a non-constant coefficient function. Using a variational approach we prove new regularity properties of a weak solution whose existence and uniqueness in anisotropically weighted Sobolev spaces were proved in [10]. Because we use some facts and technical tools proved in the above mentioned paper, we give also a brief review of its results and methods.

We study the global attractor of the non-autonomous 2D Navier–Stokes system with time-dependent external force $g(x,t)$. We assume that $g(x,t)$ is a translation compact function and the corresponding Grashof number is small. Then the global attractor has a simple structure: it is the closure of all the values of the unique bounded complete trajectory of the Navier–Stokes system. In particular, if $g(x,t)$ is a quasiperiodic function with respect to $t$, then the attractor is a continuous image of a torus. Moreover the...

We study the global attractor of the non-autonomous 2D Navier–Stokes system with time-dependent external force g(x,t). We assume that g(x,t) is a translation compact function and the corresponding Grashof number is small. Then the global attractor has a simple structure: it is the closure of all the values of the unique bounded complete trajectory of the Navier–Stokes system. In particular, if g(x,t) is a quasiperiodic function with respect to t, then the attractor is a continuous image...

We consider the $L^2$-critical focusing non-linear Schrödinger equation in $1+1$-d. We demonstrate the existence of a large set of initial data close to the ground state soliton resulting in the pseudo-conformal type blow-up behavior. More specifically, we prove a version of a conjecture of Perelman, establishing the existence of a codimension one stable blow-up manifold in the measurable category.

The existence of stationary solutions and blow up of solutions for a system describing the interaction of gravitationally attracting particles that obey the Fermi-Dirac statistics are studied.

The present paper describes mobile carrier transport in semiconductor devices with constant densities of ionized impurities. For this purpose we use one-dimensional partial differential equations. The work gives the proofs of global existence of solutions of systems of such kind, their bifurcations and their stability under the corresponding assumptions.

We study existence and approximation of non-negative solutions of partial differential equations of the type$${\partial }_{t}u-div\left(A(\nabla \left(f\left(u\right)\right)+u\nabla V)\right)=0\text{in}(0,+\infty )\times {ℝ}^n,(0.1)$$where $A$ is a symmetric matrix-valued function of the spatial variable satisfying a uniform ellipticity condition, $f:[0,+\infty )\to [0,+\infty )$ is a suitable non decreasing function, $V:{ℝ}^n\to ℝ$ is a convex function. Introducing the energy functional $\phi \left(u\right)={\int }_{{ℝ}^n}F\left(u\left(x\right)\right)\mathrm{d}x+{\int }_{{ℝ}^n}V\left(x\right)u\left(x\right)\mathrm{d}x$, where $F$ is a convex function linked to $f$ by $f\left(u\right)=u{F}^{\text{'}}\left(u\right)-F\left(u\right)$, we show that $u$ is the “gradient flow” of $\phi$ with respect to the 2-Wasserstein distance between probability measures on the space...

We study existence and approximation of non-negative solutions of partial differential equations of the type 
 $${\partial }_{t}u-div\left(A(\nabla \left(f\left(u\right)\right)+u\nabla V)\right)=0\text{in}(0,+\infty )\times {ℝ}^n,(0.1)$$ where A is a symmetric matrix-valued function of the spatial variable satisfying a uniform ellipticity condition, $f:[0,+\infty )\to [0,+\infty )$ is a suitable non decreasing function, $V:{ℝ}^n\to ℝ$ is a convex function. Introducing the energy functional $\phi \left(u\right)={\int }_{{ℝ}^n}F\left(u\left(x\right)\right)\mathrm{d}x+{\int }_{{ℝ}^n}V\left(x\right)u\left(x\right)\mathrm{d}x$, where F is a convex function linked to f by $f\left(u\right)=u{F}^{\text{'}}\left(u\right)-F\left(u\right)$, we show that u is the “gradient flow” of ϕ with respect to the 2-Wasserstein distance between probability measures on the space...

The paper is devoted to the analysis of an abstract evolution inclusion with a non-invertible operator, motivated by problems arising in nonlocal phase separation modeling. Existence, uniqueness, and long-time behaviour of the solution to the related Cauchy problem are discussed in detail.