Sul comportamento asintotico delle soluzioni di equazioni non lineari di tipo parabolico. Nota I

Vaghi, Carla

Atti della Accademia nazionale dei Lincei. Rendiconti della Classe di scienze fisiche, matematiche e naturali, Série 8, Tome 41 (1966) no. 1-2, pp. 25-31.

Résumé

Consider the non linear parabolic equation \begin{equation} \tag{I} \frac{\partial z(x,t)}{\partial t} = \sum_{i,j}^{1 \dots m} \, a_{ij}(x,t) \, \frac{\partial^{2} z(x,t)}{\partial x_{i}\partial x_{j}} - f(x,t,z,p) \end{equation}\begin{equation*} \left( x = x_{1}, \cdots, x_{m} \in \Omega \,\, ; \,\, p = p_{1}, \cdots, p_{m} \,\, ; \,\, p_{i} = \frac{\partial z}{\partial x_{i}} \right), \end{equation*} with the boundary condition $z(x,t)|_{\partial \Omega} = 0$. Under appropriate assumptions on the function $f(x,t,z,p)$, it is proved that all the solutions of (I) have the same asymptotic behaviour when $t \to +\infty$. Subsequently, some uniqueness theorems of solutions bounded on $(-\infty, +\infty)$ are given.