Existence Theorems for a Quasilinear Evolution Equation
John Clements · SIAM Journal on Applied Mathematics · 1974
Sufficient conditions on the functions $a_i ,i = 1, \cdots ,N$, and f are determined which ensure, via a monotonicity argument, the existence of weak periodic solutions of the Dirichlet problem for \[ u_{tt} - \frac{d} {{dx_i }}a_i (x,t,u_{xi} ) - {\bf \Delta } _N \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\cdot}$}}{u} _t = f,\qquad ,(x,t) \in \Omega \times E^1 , \] where ${\bf \Omega }$ is a bounded domain in $E^N $. In addition to possessing some smoothness, each $a_i $ is required to satisfy \[ \begin{gathered} \left| {a_i (x,t,\eta )} \right| \leqq k_2 \left\{ | \eta |^{p - 1} + 1\right\} \quad ({\text{boundedness}}) \hfill \\ a_i (x,t,\eta )\gtrless k_0 | \eta |^{p - 2} \eta ,\quad \eta \gtrless 0,\quad ({\text{coercivity}}) \hfill \\ (\partial/\partial t)a_i (x,t,\eta )\gtrless 0,\quad \eta \lessgtr 0, \hfill \\ (a_i (x,t,\eta ) - a_i (x,t,\xi ))(\eta - \xi )\geqq 0\quad ({\text{monotonicity}}) \hfill \\ \end{gathered} \] for some $p\geqq 2$, some constants $k_0 $ and $k_2 $ and all real $\xi $ and $\eta $. It is also shown that these conditions ensure the existence of weak global solutions of initial-boundary value problems for this equation.