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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 \(L^ \infty (\mathbb R^ +; L^ 1_{\zeta\mu} (M))\).

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
Full Text: DOI
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