Uniqueness theorem for $p$-biharmonic equations

Benedikt, Jiří

Electronic Journal of Differential Equations, Tome 2002 (2002).

Résumé

Summary: The goal of this paper is to prove existence and uniqueness of a solution of the initial value problem for the equation $$ (|u''|^{p-2}u'')''=\lambda |u|^{q-2}u $$ where $\lambda\in{\mathbb{R}}$ and $p,q>1$. We prove the existence for $p\geq q$ only, and give a counterexample which shows that for $p$ there need not exist a global solution (blow-up of the solution can occur). On the other hand, we prove the uniqueness for $p\leq q$, and show that for $p>q$ the uniqueness does not hold true (we give a corresponding counterexample again). Moreover, we deal with continuous dependence of the solution on the initial conditions and parameters.

Détail

Classification : 34A12, 34C11, 34L30
Keywords: p-biharmonic operator, existence and uniqueness of solution, continuous dependence on initial conditions, jumping nonlinearity