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Integral representation of solutions of semilinear elliptic equations in cylinders and applications. (English) Zbl 0817.35022
From the introduction: Let $$(M,\mu)$$ be a positive measured space and $$L$$ a linear operator of $$L^ 1_ \mu (M)$$ with dense domain $$\text{Dom} (L)$$. We consider the following equation in $$\mathbb R^ + \times M$$ $\partial_{tt} u+a \partial_ t u+ Lu-lu+ f(u)+ \Phi(t)=0 \tag{1}$ and the following hypotheses: $$f$$ is a nonnegative locally Lipschitz continuous function defined on $$\mathbb R^ +$$ vanishing at 0 and satisfying $$\lim f(u)/u= \infty$$ (as $$u\to +\infty)$$, $$L$$ is a linear $$T$$-dissipative operator of $$L^ 1_ \mu (M)$$ with domain $$\text{Dom} (L)$$ and there exists $$\zeta\in L^ 1_ \mu (M)\cap L^ \infty_ \mu (M)$$ such that $$\zeta\geq 0$$, $$\int_ M \zeta \,d\mu =1$$ and $\int_ M \zeta L\omega \,d\mu\geq- \lambda \int_ M \int_ M \zeta\omega \,d\mu \qquad (\forall\omega\in \text{Dom} (L),\;\omega\geq 0 \text{ a.e.})$ for some constant $$\lambda\geq 0$$. We assume, moreover, that $$L$$ is $$m$$-dissipative with dense domain in $$L^ 1_{\zeta\mu} (M)$$, where $$L^ 1_{\zeta\mu} (M)$$ is the space of $$\mu$$-measurable functions which are integrable for the measure $$\zeta\mu$$. $$\Phi$$ belongs to $$L^ 1_{\text{loc}} (\mathbb R^ +; L^ 1 (M))$$, $$\Phi\geq 0$$ and $$a$$ and $$l$$ are constants, $$l>0$$.
We call $$(S(t) )_{t\geq 0}$$ the continuous semigroup of sub-Markovian operators of $$L^ 1_{\zeta \mu} (M)$$ generated by $$-(-L+ (a^ 2/4+ l)I)^{1/2}$$. Our main result is the following theorem.
Theorem. Let $$u$$ belong to $$W^{2,1}_{\text{loc}} ([0, \infty); (L^ 1_ \mu (M)))\cap L^ 1_{\text{loc}} ([0, \infty); \text{Dom} (L))$$ be a solution of (1) such that $$u\geq 0$$ a.e. on $$\mathbb R^ + \times M$$. Then $$\int_ \rho^{\rho+1} \int_ M (f(u)+ \Phi) (t) \zeta \,d\mu \,dt$$ remain bounded independently of $$\rho\geq 0$$ and the following formula is valid $u(t)= e^{-at/2} S(t) u(0)+ \int_ 0^ t e^{-as/2} S(s) \int_ 0^ \infty e^{a\tau/2} S(\tau) (f(u)+ \Phi)(t+ \tau- s)\,d\tau \, ds$ for any $$t\geq 0$$. Therefore, $$u$$ belongs to $$L^ \infty (\mathbb R^ +; L^ 1_{\zeta\mu} (M))$$.

##### MSC:
 35J61 Semilinear elliptic equations 35C15 Integral representations of solutions to PDEs 58J05 Elliptic equations on manifolds, general theory 35B45 A priori estimates in context of PDEs 47D07 Markov semigroups and applications to diffusion processes
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