In this work, using bilinear estimates in Bourgain type spaces, we prove the local existence of a solution to a higher order nonlinear dispersive equation on the line for analytic initial data $u_0$. The analytic initial data can be extended as holomorphic functions in a strip around the $x$-axis. By Gevrey approximate conservation law, we prove the existence of the global solutions, which improve earlier results of Z. Zhang, Z. Liu, M. Sun, S. Li, (2019).

In this paper we construct a minimizing sequence for the problem (1). In particular, we show that for any subsolution of the Hamilton-Jacobi equation $(*)$ there exists a minimizing sequence weakly convergent to this subsolution. The variational problem (1) arises from the theory of computer vision equations.

The paper is devoted to infinitely differentiable one-parameter convolution semigroups in the convolution algebra ${}_C^{\text{'}}(ℝⁿ;M_{m\times m})$ of matrix valued rapidly decreasing distributions on ℝⁿ. It is proved that $G{\in }_C^{\text{'}}(ℝⁿ;M_{m\times m})$ is the generating distribution of an i.d.c.s. if and only if the operator ${\partial }_t{\otimes }_{m\times m}-G\ast$ on ${ℝ}^{1+n}$ satisfies the Petrovskiĭ condition for forward evolution. Some consequences are discussed.

On montre l’équivalence entre certaines inégalités “à la Carleman” et certaines propriétés de régularité des solutions à support compact d’équations aux dérivées partielles à coefficients constants : $P\left(D\right)$ étant un opérateur différentiel sur ${\mathbf{R}}^n$, on en déduit une caractérisation, en termes d’inégalités $L^2$, des ouverts $\Omega$ de ${\mathbf{R}}^n$ tels que $\Omega \times {\mathbf{R}}^k$ soit $P\left(D\right)$-convexe pour tout entier $k$.