# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
  Aviles, P., Local behaviour of solutions of some elliptic equations, Communs math. phys., 108, 177-192, (1987) · Zbl 0617.35040  Bidaut-Veron, M.F.; Veron, L., Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations, Invent. math., 106, 489-539, (1991) · Zbl 0755.35036  Gilgarg, D.; Trudinger, N.S., Elliptic partial differential equations of second order, grundleheren math. wiss., Vol. 224, (1983), Springer Berlin  Gidas, B.; Spruck, J., Global and local behaviour of positive solutions of nonlinear elliptic equations, Communs pure appl. math., 34, 525-598, (1981) · Zbl 0465.35003  Bidaut-Veron, M.F.; Bouhar, M., On characterization of solutions on some nonlinear differential equations and applications, SIAM J. math. analysis, 25, 859-875, (1994) · Zbl 0807.34050  Balakrishnan, I.V., Fractional powers of closed operators and the semi-groups generated by them, Pacif. J. math., 10, 419-437, (1961) · Zbl 0103.33502  Brezis, H.; Strauss, W.A., Semilinear second order elliptic equation in L1, J. math. soc. Japan, 25, 565-590, (1973) · Zbl 0278.35041  Bandle, C.; Essen, M., On the positive solutions of nonlinear elliptic equations in cone-like domains, Archs ration. mech. analysis, 112, 319-338, (1990) · Zbl 0727.35051  Bardos, C., Problèmes aux limites pour des équations aux dérivées partielles du premier ordre à coefficients réels, Ann. sci. E.N.S., 3, 185-233, (1970) · Zbl 0202.36903  Veron, L., Equations d’évolution semi-linéaires du second ordre dans L1, Rev. roum. math. pura appl., 27, 95-123, (1982) · Zbl 0489.35010  Veron, L., Comportement asymptotique des solutions d’équations elliptiques semi-linéaires dans $$R$$^N, Annali mat. pura appl., 127, 25-50, (1981) · Zbl 0467.35013  Stein, E.M., Boundary behaviour of holomorphic functions of several complex variables, (1972), Princeton University Press Princeton, (Mathematics Notes No. 9)  Stein, E.M.; Weiss, G., On the convergence of Poisson integrals, Trans. am. math. soc., 140, 35-54, (1969) · Zbl 0182.10801
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.