Solutions of Smooth Nonlinear Partial Differential Equations

and Applied Analysis 3 definition, it may happen that such a solution does not belong to any of the customary spaces of generalized functions. For example, given a function u : C \ {z0} −→ C 1.3 which is analytic everywhere except at the single point z0 ∈ C, and with an essential singularity at z0, Picard’s Theorem states that u attains every complex value, with possibly one exception, in every neighborhood of z0. Clearly such a function does not satisfy any of the usual growth conditions that are, rather as a rule, imposed on generalized functions. Indeed, we may recall that the elements of a Sobolev space are locally integrable, while the elements of the Colombeau algebras 14 , which contain the D′ distributions, must satisfy certain polynomial type growth conditions near singularities. Therefore these concepts of generalized functions cannot accommodate thementioned singularity of the function in 1.3 . In this paper, we present further developments of the general and type-independent solution method presented in 1 , and in particular the uniform convergence spaces of generalized functions introduced in 4–6 . Furthermore, and in contradistinction with the spaces of generalized functions introduced in 6 , we construct here a space of generalized functions that admit generalized partial derivatives of arbitrary order. While, following the methods introduced in 6 , one may easily construct such a space of generalized functions, the existence of generalized solutions of systems of nonlinear PDEs in this space is nontrivial. Here we present the mentioned construction of the space of generalized functions, and show how generalized solutions of a large class of C∞-smooth nonlinear PDEs may be obtained in this space. As an application of the general theory, we discuss also the existence and regularity of generalized solutions of the parametrically driven, damped nonlinear Schrödinger equation in one spatial dimension. In this regard, we show that for a large class of C∞-smooth initial values, the mentioned Schrödinger equation admits a generalized solution that satisfies the initial condition in a suitable generalized sense. We also introduce the concept of a strongly generic weak solution of this equation, and show that the solution we construct is such a weak solution. The paper is organized as follows. In Section 2 we recall some basic facts concerning normal lower semicontinuous functions from the literature. The construction of spaces of generalized functions is given in Section 3, while Section 4 is concerned with the existence of generalized solutions ofC∞-smooth nonlinear PDEs. Lastly, in Section 5, we apply the general method to the parametrically driven, damped nonlinear Schrödinger equation in one spatial dimension. For all details on convergence spaces we refer the reader to the excellent book 15 and the paper 16 . 2. Normal Lower Semicontinuous Functions The concept of a normal lower semicontinuous function was first introduced by Dilworth 17 in connection with his attempts at characterizing the Dedekind order completion of spaces of continuous functions, a problem that was solved only recently by Anguelov 3 . Here we recall some facts concerning normal semicontinuous functions. For more details, and the proofs of some of the results, we refer the reader to the more recent presentations in 4, 18 . 4 Abstract and Applied Analysis Denote byR R∪{±∞} the extended real line, ordered as usual. The set of all extended real-valued functions on a topological space X is denotedA X . A function u ∈ A X is said to be nearly finitewhenever {x ∈ X : u x ∈ R} is open and dense. 2.1 Two fundamental operations on the space A X are the Lower and Upper Baire Operators I : A X −→ A X , S : A X −→ A X , 2.2 introduced by Baire 19 , see also 3 , which are defined by I u : X x −→ supinfuy : y ∈ V } : V ∈ Vx } , 2.3 S u : X x −→ infsupuy : y ∈ V } : V ∈ Vx } , 2.4 respectively, with Vx denoting the neighborhood filter at x ∈ X. Clearly, the Baire operators I and S satisfy I u ≤ u ≤ S u , u ∈ A X , 2.5 whenA X is equipped with the usual pointwise order u ≤ v ⇐⇒ ∀x ∈ X : u x ≤ v x . 2.6 Furthermore, the Baire operators, as well as their compositions, are idempotent and monotone with respect to the pointwise order. That is, ∀u ∈ A Ω : 1 I I u I u , 2 S S u S u , 3 I ◦ S I ◦ S u I ◦ S u , ∀u, v ∈ A Ω : u ≤ v ⇒ ⎛ ⎝ 1 I u ≤ I v 2 S u ≤ S v 3 I ◦ S u ≤ I ◦ S v ⎞ ⎠. 2.7 The operators I and S, as well as their compositions I ◦ S and S ◦ I, are useful tools for the study of extended real-valued functions. In this regard, we may mention that these Abstract and Applied Analysis 5 mappings characterize certain continuity properties of functions in A X . In particular, we haveand Applied Analysis 5 mappings characterize certain continuity properties of functions in A X . In particular, we have u ∈ A X is lower semicontinuous ⇐⇒ I u u, u ∈ A X is upper semicontinuous ⇐⇒ S u u. 2.8 Furthermore, a function u ∈ A X is normal lower semicontinuous on X whenever I ◦ S u u. 2.9 We denote the set of nearly finite normal lower semicontinuous functions on X by NL X . The concept of normal lower semicontinuity of extended real-valued functions extends that of continuity of usual real-valued functions. In particular, each continuous function is nearly finite and normal lower semicontinuous so that we have the inclusion