We investigate stationary solutions and asymptotic behaviour of solutions of two boundary value problems for semilinear parabolic equations. These equations involve both blow up and damping terms and they were studied by several authors. Our main goal is to fill some gaps in these studies.

Global-in-time existence of solutions for equations of viscous compressible barotropic fluid in a bounded domain Ω ⊂ ${ℝ}^3$ with the boundary slip condition is proved. The solution is close to an equilibrium solution. The proof is based on the energy method. Moreover, in the $L_2$-approach the result is sharp (the regularity of the solution cannot be decreased) because the velocity belongs to $H^{2+\alpha ,1+\alpha /2}\left(\Omega \times {ℝ}_{+}\right)$ and the density belongs to $H^{1+\alpha ,1/2+\alpha /2}\left(\Omega \times {ℝ}_{+}\right)$, α ∈ (1/2,1).

Global existence of regular solutions to the Navier-Stokes equations for velocity and pressure coupled with the heat convection equation for temperature in a cylindrical pipe is shown. We assume the slip boundary conditions for velocity and the Neumann condition for temperature. First we prove long time existence of regular solutions in [kT,(k+1)T]. Having T sufficiently large and imposing some decay estimates on ${\left|\right|f\left(t\right)\left|\right|}_{L₂\left(\Omega \right)}$, $\left|\right|f_{,x₃}\left(t\right){\left|\right|}_{L₂\left(\Omega \right)}$ we continue the local solutions step by step up to a global one.

We prove that the 3D cubic defocusing semi-linear wave equation is globally well-posed for data in the Sobolev space Hs where s > 3/4. This result was obtained in [11] following Bourgain's method ([3]). We present here a different and somewhat simpler argument, inspired by previous work on the Navier-Stokes equations ([4, 7]).

We establish the global existence and uniqueness of smooth solutions to a nonlinear Alfvén wave equation arising in a finite-beta plasma. In addition, the spatial asymptotic behavior of the solution is discussed.

A hybrid flexible beam equation with harmonic disturbance at the end where a rigid tip body is attached is considered. A simple motor torque feedback control is designed for which only the measured time-dependent angle of rotation and its velocity are utilized. It is shown that this control can impel the amplitude of the attached rigid tip body tending to zero as time goes to infinity.

A hybrid flexible beam equation with harmonic disturbance at the end where a rigid tip body is attached is considered. A simple motor torque feedback control is designed for which only the measured time-dependent angle of rotation and its velocity are utilized. It is shown that this control can impel the amplitude of the attached rigid tip body tending to zero as time goes to infinity.

We consider $2m$th-order $(m\ge 2)$ semilinear parabolic equations $u_t=-{\left(-\Delta \right)}^{m}u\pm {\left|u\right|}^{p-1}u$ in ${ℛ}^N\times {ℝ}_{+}$$(p>1)$, with Dirac’s mass $\delta \left(x\right)$ as the initial function. We show that for $p<p_0=1+2m/N$, the Cauchy problem admits...

This paper is concerned with strong solutions of uniformly elliptic equations of non-divergence type in the plane. First, we use the notion of quasiregular gradient mappings to improve Morrey’s theorem on the Hölder continuity of gradients of solutions. Then we show that the Gilbarg-Serrin equation does not produce the optimal Hölder exponent in the considered class of equations. Finally, we propose a conjecture for the best possible exponent and prove it under an additional restriction.

We discuss the interior Hölder everywhere regularity for minimizers of quasilinear functionals of the type $$𝒜(u;\Omega )={\int }_{\Omega }{A}_{ij}^{\alpha \beta }(x,u)D_{\alpha }{u}^i{D}_{\beta }{u}^j\mathrm{d}x$$ whose gradients belong to the Morrey space $L^{2,n-2}(\Omega ,{ℝ}^{nN})$.

In this paper the problem of homogeneization for the Laplace operator in partially perforated domains with small cavities and the Neumann boundary conditions on the boundary of cavities is studied. The corresponding spectral problem is also considered.