Let E be a Banach space. We consider a Cauchy problem of the type ⎧ $D_t^{k}u+{\sum }_{j=0}^{k-1}{\sum }_{\left|\alpha \right|\le m}{A}_{j,\alpha }\left(D_t^j{D}_x^{\alpha }u\right)=f$ in ${ℝ}^{n+1}$, ⎨ ⎩ $D_t^{j}u(0,x)={\phi }_j\left(x\right)$ in ${ℝ}^n$, j=0,...,k-1, where each $A_{j,\alpha }$ is a given continuous linear operator from E into itself. We prove that if the operators $A_{j,\alpha }$ are nilpotent and pairwise commuting, then the problem is well-posed in the space of all functions $u\in C^{\infty }({ℝ}^{n+1},E)$ whose derivatives are equi-bounded on each bounded subset of ${ℝ}^{n+1}$.

In this article, we prove the partial exact controllability of a nonlinear system. We use semigroup formulation together with fixed point approach to study the nonlinear system.


We prove a $C^{k,\alpha }$ partial regularity result for local minimizers of variational integrals of the type $I\left(u\right)={\int }_{\Omega }f\left(D^{k}u\left(x\right)\right)\mathrm{d}x$, assuming that the integrand f satisfies (p,q) growth conditions.


This paper is concerned with periodic solutions for perturbations of the sweeping process introduced by J.J. Moreau in 1971. The perturbed equation has the form $-Du\in N_{C\left(t\right)}\left(u\left(t\right)\right)+f(t,u\left(t\right))$ where C is a T-periodic multifunction from [0,T] into the set of nonempty convex weakly compact subsets of a separable Hilbert space H, $N_{C\left(t\right)}\left(u\left(t\right)\right)$ is the normal cone of C(t) at u(t), f:[0,T] × H∪H is a Carathéodory function and Du is the differential measure of the periodic BV solution u. Several existence results of periodic solutions for this...

We show that the Porous Medium Equation and the Fast Diffusion Equation, $\dot{u}-\Delta u^m=f$, with $m\in (0,\infty )$, can be modeled as a gradient system in the Hilbert space $H^{-1}\left(\Omega \right)$, and we obtain existence and uniqueness of solutions in this framework. We deal with bounded and certain unbounded open sets $\Omega \subseteq {ℝ}^n$ and do not require any boundary regularity. Moreover, the approach is used to discuss the asymptotic behaviour and order preservation of solutions.

We generalize the Malgrange preparation theorem to matrix valued functions $F(t,x)\in C^{\infty }(\mathbf{R}\times {\mathbf{R}}^n)$ satisfying the condition that $t\mapsto \det F(t,0)$ vanishes to finite order at $t=0$. Then we can factor $F(t,x)=C(t,x)P(t,x)$ near (0,0), where $C(t,x)\in C^{\infty }$ is inversible and $P(t,x)$ is polynomial function of $t$ depending $C^{\infty }$ on $x$. The preparation is (essentially) unique, up to functions vanishing to infinite order at $x=0$, if we impose some additional conditions on $P(t,x)$. We also have a generalization of the division theorem, and analytic versions generalizing the Weierstrass preparation...