The Cauchy problem for the higher order equations in the mKdV hierarchy is investigated with data in the spaces $${\widehat{H}}_s^r\left(ℝ\right)$$ defined by the norm $${∥v_0∥}_{{\widehat{H}}_s^r\left(ℝ\right)}:={∥{〈\xi 〉}^s\widehat{v_0}∥}_{L_{\xi }^{r^{\text{'}}}},〈\xi 〉={\left(1+{\xi }^2\right)}^{\frac{1}{2}},\frac{1}{r}+\frac{1}{r^{\text{'}}}=1$$ . Local well-posedness for the jth equation is shown in the parameter range 2 ≥ 1, r > 1, s ≥ $$\frac{2j-1}{2r^{\text{'}}}$$ . The proof uses an appropriate variant of the Fourier restriction norm method. A counterexample is discussed to show that the Cauchy problem for equations of this type is in general ill-posed in the C 0-uniform sense, if s < $$\frac{2j-1}{2r^{\text{'}}}$$ . The results for r = 2 - so far in...

Non-linear second order parabolic systems in the divergent form are considered. It is proved that under some restrictions on the modulus of ellipticity, all weak solutions are continuous.

We prove the interior Hölder continuity of weak solutions to parabolic systems $$\frac{\partial u^j}{\partial t}-D_{\alpha }{a}_j^{\alpha }(x,t,u,\nabla u)=0\text{in}Q(j=1,...,N)$$ ($Q=\Omega \times (0,T),\Omega \subset {ℝ}^2$), where the coefficients $a_j^{\alpha }(x,t,u,\xi )$ are measurable in $x$, Hölder continuous in $t$ and Lipschitz continuous in $u$ and $\xi$.

In this paper we study the behavior of solutions of the boundary value problem for the Poisson equation in a partially perforated domain with arbitrary density of cavities and mixed type conditions on their boundary. The corresponding spectral problem is also considered. A short communication of similar results can be found in [1].

We analyse the effect of the mechanical response of the solid phase during liquid/solid phase change by numerical simulation of a benchmark test based on the well-known and debated experiment of melting of a pure gallium slab counducted by Gau & Viskanta in 1986. The adopted mathematical model includes the description of the melt flow and of the solid phase deformations. Surprisingly the conclusion reached is that, even in this case of pure material, the contribution of the solid phase to the...

We demonstrate a theorem of existence and uniqueness on a large scale of the solution of a system of differential disequations associated to a Graffi model relative to the motion of two incompressible viscous fluids.

In this paper, we propose a computational model to investigate the coupling between cell’s adhesions and actin fibres and how this coupling affects cell shape and stability. To accomplish that, we take into account the successive stages of adhesion maturation from adhesion precursors to focal complexes and ultimately to focal adhesions, as well as the actin fibres evolution from growing filaments, to bundles and finally contractile stress fibres.We use substrates with discrete patterns of adhesive...