We are concerned with the problem of differentiability of the derivatives of order $m+1$ of solutions to the “nonlinear basic systems” of the type $${(-1)}^m\sum _{\left|\alpha \right|=m}{D}^{\alpha }{A}^{\alpha }\left(D^{\left(m\right)}u\right)+\frac{\partial u}{\partial t}=0\text{in}Q.$$ We are able to show that $$D^{\alpha }u\in L^2(-a,0,H^{\vartheta }(B\left(\sigma \right),{ℝ}^N)),|\alpha |=m+1,$$ for $\vartheta \in (0,1/2)$ and this result suggests that more regularity is not expectable.

We consider the motion of a viscous compressible heat conducting fluid in ℝ³ bounded by a free surface which is under constant exterior pressure. Assuming that the initial velocity is sufficiently small, the initial density and the initial temperature are close to constants, the external force, the heat sources and the heat flow vanish, we prove the existence of global-in-time solutions which satisfy, at any moment of time, the properties prescribed at the initial moment.

Consider a stochastic heat equation ∂tu=κ  ∂xx2u+σ(u)ẇ for a space–time white noise ẇ and a constant κ>0. Under some suitable conditions on the initial function u0 and σ, we show that the quantities lim sup t→∞t−1sup x∈Rln El(|ut(x)|2) and lim sup t→∞t−1ln E(sup x∈R|ut(x)|2) are equal, as well as bounded away from zero and infinity by explicit multiples of 1/κ. Our proof works by demonstrating quantitatively that the peaks of the stochastic process x↦ut(x) are highly concentrated...

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...

A theorem on the existence of weak solutions of the Cauchy problem for first order functional differential equations defined on the Haar pyramid is proved. The initial problem is transformed into a system of functional integral equations for the unknown function and for its partial derivatives with respect to spatial variables. The method of bicharacteristics and integral inequalities are applied. Differential equations with deviated variables and differential integral equations can be obtained...

We consider the local initial value problem for the hyperbolic partial functional differential equation of the first order (1) $Dₓz(x,y)=f(x,y,z(x,y),\left(Wz\right)(x,y),D_{y}z(x,y))$ on E, (2) z(x,y) = ϕ(x,y) on [-τ₀,0]×[-b,b], where E is the Haar pyramid and τ₀ ∈ ℝ₊, b = (b₁,...,bₙ) ∈ ℝⁿ₊. Using the method of bicharacteristics and the method of successive approximations for a certain functional integral system we prove, under suitable assumptions, a theorem on the local existence of weak solutions of the problem (1),(2).

In this article we prove new results concerning the structure and the stability properties of the global attractor associated with a class of nonlinear parabolic stochastic partial differential equations driven by a standard multidimensional brownian motion. We first use monotonicity methods to prove that the random fields either stabilize exponentially rapidly with probability one around one of the two equilibrium states, or that they set out to oscillate between them. In the first case we can...

In this article we prove new results concerning the structure and the stability properties of the global attractor associated with a class of nonlinear parabolic stochastic partial differential equations driven by a standard multidimensional Brownian motion. We first use monotonicity methods to prove that the random fields either stabilize exponentially rapidly with probability one around one of the two equilibrium states, or that they set out to oscillate between them. In the first case we can...

We consider a heat equation with a non-linear right-hand side which depends on certain Volterra-type functionals. We study the problem of existence and convergence for the method of lines by means of semi-discrete inverse formulae.

We consider the mixed problem for the hyperbolic partial differential-functional equation of the first order $$D_{x}z(x,y)=f(x,y,z_{(x,y)},D_{y}z(x,y)),$$ where $z_{(x,y)}[-\tau ,0]\times [0,h]\to ℝ$ is a function defined by $z_{(x,y)}(t,s)=z(x+t,y+s)$, $(t,s)\in [-\tau ,0]\times [0,h]$. Using the method of bicharacteristics and the method of successive approximations for a certain integral-functional system we prove, under suitable assumptions, a theorem of the local existence of generalized solutions of this problem.

We consider the mixed problem for the quasilinear partial functional differential equation with unbounded delay $D_{t}z(t,x)={\sum }_{i=1}^n{f}_i(t,x,z_{(t,x)})D_{x_i}z(t,x)+h(t,x,z_{(t,x)})$, where $z_{(t,x)}\in X{̶}_0$ is defined by $z_{(t,x)}(\tau ,s)=z(t+\tau ,x+s)$, $(\tau ,s)\in (-\infty ,0]\times [0,r]$, and the phase space $X{̶}_0$ satisfies suitable axioms. Using the method of bicharacteristics and the fixed-point method we prove a theorem on the local existence and uniqueness of Carathéodory solutions of the mixed problem.

We study the flat region of stationary points of the functional ${\int }_{\Omega }F\left(\right|\nabla u\left(x\right)\left|\right)dx$ under the constraint $u\le M$, where $\Omega$ is a bounded domain in ${ℝ}^2$. Here $F\left(s\right)$ is a function which is concave for $s$ small and convex for $s$ large, and $M>0$ is a given constant. The problem generalizes the classical minimal resistance body problems considered by Newton. We construct a family of partially flat radial solutions to the associated stationary problem when $\Omega$ is a ball. We also analyze some other qualitative properties. Moreover, we show the...