# zbMATH — the first resource for mathematics

Evolution equations governed by families of weighted operators. (English) Zbl 0926.34051
A Lebesgue approach for a fully nonlinear nonautonomous evolution problem in an arbitrary Banach space $$X$$ $\frac{du}{dt}+\alpha(t)A_\alpha(t)u\ni 0, \quad t\in I\subseteq[0,T], \quad u(s)=u^0, \tag{1}$ is developed. (Here, the control $$\alpha$$ acts on an unbounded operator). For this purpose an abstract $$L^1$$-comparison mode (called coherence) between multivalued time dependent families of operators $$(A_\alpha(s))_{s\in I}$$ and $$(A_\beta(t))_{t\in J}$$ on compact subintervals $$I, J \subseteq[0,T]$$ weighted by functions $$\alpha, \beta\in L^\infty([0,T], \mathbb{R}^+)$$ is defined. The solution to the Cauchy problem (1) called mas is given as a limit of discrete implicit schemes, where approximations are in a Lebesgue sense. The main results extending Crandall’s, Ligett’s, Evans’, and others’are
(1) existence and uniqueness results: all $$\varepsilon_n$$-discrete adapted approximating families (DAF) to (1) are uniformly convergent on $$I$$ towards its unique mas;
(2) estimates for coherent mases are obtained;
(3) $$S(t,s)$$ defined as $$S(t,s)u^0=u(t)$$ is proved to be an evolution operator for a strongly $$\psi$$-coherent family $$(A_\alpha)_{[0,T]}$$.

##### MSC:
 34G20 Nonlinear differential equations in abstract spaces 35G10 Initial value problems for linear higher-order PDEs 35K25 Higher-order parabolic equations 47J05 Equations involving nonlinear operators (general) 47N40 Applications of operator theory in numerical analysis
Full Text:
##### References:
 [1] Bénilan, P., Équations d’évolution dans un espace de Banach quelconque et applications, Thèse, Paris-XI, orsay, (1972) [2] Ph. , M.G. \scCrandall and A. \scPazy, Evolution equations governed by accretive operators, Book (to appear). [3] Ball, J.M.; Marsden, J.E.; Slemrod, M., Controllability of distributed bilinear systems, SIAM J. control optim., Vol. 20, 6, 575-597, (1982) · Zbl 0485.93015 [4] Brezis, H., Analyse fonctionnelle, () · Zbl 0511.46001 [5] Couchouron, J.-F., Équations d’évolution: le problème de Cauchy, (1993), Univ. de Rouen Rouen, These [6] J.-F. \scCouchouron, Problème de Cauchy non autonome pour des équations d’évolution, à paraitre. [7] J.-F. \scCouchouron and P. \scLigarius, Weighted evolution equations and asymptotic observers in Banach spaces. A nonlinear approach, submitted. [8] Crandall, M.G., Nonlinear semigroups and evolution governed by accretive operators, (), 305-337, (Part. I) · Zbl 0637.47039 [9] Crandall, M.G.; Evans, L.C., On the relation of the operator $$(∂∂s) + (∂∂τ)$$ to evolution governed by accretive operators, Israël J. math., Vol. 21, 4, 261-278, (1975) · Zbl 0351.34037 [10] Crandall, M.G.; Liggett, T.M., Generation of semigroups of nonlinear transformations on general Banach spaces, American J. math., Vol. 93, 265-298, (1971) · Zbl 0226.47038 [11] Diestel, J.; Uhl, J., Vector measure, Math. surveys - AMS, Vol. 15, (1977) · Zbl 0369.46039 [12] Evans, L.C., Nonlinear evolution equations in an arbitrary Banach space, Israël J. math., Vol. 26, 1, 1-42, (1977) · Zbl 0349.34043 [13] Hartman, Ordinary differential equations, (1964), J. Wiley & Sons · Zbl 0125.32102 [14] Hinrichsen, D.; Pritchard, A.J., Robust stability of linear evolution operators on Banach spaces, SIAM J. control optim., Vol. 32, 6, 1503-1541, (1994) · Zbl 0817.93055 [15] Kobayasi, K.; Kobayashi, Y.; Oharu, S., Nonlinear evolution operators in Banach spaces, Osaka J. math., Vol. 21, 281-310, (1984) · Zbl 0567.47047 [16] Ligarius, P., Observateurs de systemes bilinéaires a parametres répartis - applications a un échangeur thermique, () [17] Lakshmikantham, V.; Leela, S., Nonlinear differential equations in abstract spaces, (1981), Pergamon · Zbl 0456.34002 [18] Tanaka, N.; Kobayashi, K., Nonlinear semigroups and evolution equations governed by generalized dissipative operators, Adv. math. sci. appl., Tokyo, Vol. 3, 401-426, (1994) · Zbl 0818.34033 [19] Walter, W., Differential and integral inequalities, (1970), Springer-Verlag [20] Xu, C.Z., Exact observability and exponential stability of infinite dimensional bilinear systems, Math. control signal systems, Vol. 9, 1, 73-93, (1996) · Zbl 0862.93007
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.