On nonlinear evolution equation of second order in Banach spaces

Here we study the existence of a solution and also the behavior of the existing solution of the abstract nonlinear di erential equation of second order that, in particular, is the nonlinear hyperbolic equation with nonlinear main parts, and in the special case, is the equation of the type of equation of tra c ow.


Introduction
In this article we study the following nonlinear evolution equation under the initial conditions where A is a linear operator in a real Hilbert space H, F ∶ X → X * and g ∶ D (g) ⊆ H × H → H are a nonlinear operators, X is a real Banach space. For example, operator A denotes −∆ with Dirichlet boundary conditions and F (u) = u ρ u (see, Example in Section 2), that in the one space dimension case, we can formulate in the form where u (x), u (x) are known functions, f (⋅) , g (⋅) ∶ R → R are continouos functions and l > is a number. The equation of type (3) describes a mathematical model of the problem from the theory of the ow in networks as is a rmed in articles [1 -4] (e. g. Aw-Rascle equations, Antman-Cosserat model, etc.). As it is noted in the survey [2], such a study can nd application in accelerating missiles and space crafts, components of high-speed machinery, manipulator arm, microelectronic mechanical structures, components of bridges and other structural elements. Balance laws are hyperbolic partial di erential equations that are commonly used to express the fundamental dynamics of open conservative systems (e.g. [3]). As the survey [2] presents su ciently exact explanations of the signi cance of equations of such type, we not discuss this theme.
We would like to note only the following physical interpretation (see, [5]): "Let V be the smooth elastic body and F be the force acting on V through ∂V with the mass density is unit. Newton's law asserts the mass times the acceleration equal the net force where u is the displacement in some direction of the point x at time t ≥ and F is a function of the displacement gradient ∇u; whence u tt + div F (∇u) = ." As it is well-known, F is a nonlinear function but for the study of this equation one usually uses a local linear approximation of F. Unlike the above mentioned works, one can study some variant of this equation with the nonlinear function F by use of the general result of this article. In this article we use di erent approach to study proposed problem that allows us to investigate the case when the main part of the problem actually contains the nonlinear operator. As is shown in the above mentioned examples, one can investigate the nonlinear hyperbolic equations with the use of the results of this article, which haven't been studied earlier. We will note that in this approach we used the Galerkin approximation method.
This article is organized as follows. In Section 2, we study the solvability of the nonlinear equation of second order in the Banach spaces, for which we found the su cient conditions and proved the existence theorem. In Section 3, we investigate the global behavior of solutions of the posed problem.

Solvability of problem (1)-(2)
Let A be a symmetric linear operator densely de ned in a real Hilbert space H and positive, A has a selfadjoint extension. Moreover, there is linear operator B de ned in H such that A ≡ B * ○ B, here f ∶ R → R is continuous as function, X is a real re exive Banach space and X ⊂ H, g ∶ D (g) ⊆ H × H → H, where g ∶ R → R is a continuous as function and x ∶ [ , T) → X is an unknown function. Let F (r) as a function be de ned as F (r) = r ∫ f (s) ds. Let the inequation x H ≤ Bx H be valid for any x ∈ D (B). We denote by V, W and by Y the spaces de ned as V ≡ {y ∈ H By ∈ H }, W = {x ∈ H Ax ∈ H } and as Y ≡ {x ∈ X Ax ∈ X }, respectively, for which inclusions W ⊂ V ⊂ H are compact and Y ⊂ W.
Let H be the real separable Hilbert space, X be the re exive Banach space and X ⊂ H ⊂ X * ; V is the previously de ned space. It is clear that W ⊂ V ⊂ H ⊂ V * ⊂ W * are framed spaces by H, these inclusions are compact and X ⊂ V * . Then one can de ne the framed spaces Since operator A is invertible, here one can set the function y (t) = A − x (t) for any t ∈ ( , T), in other words one can assume the denotation x (t) = Ay (t).
We will interpret the solution of the problem (1) -(2) in the following manner.
for any z ∈ Y and the initial conditions (2) (here and further the expression ⟨⋅, ⋅⟩ denotes the dual form for the pair: the Banach space and its dual).
Consider the following conditions (ii) Let F ∶ X → X * be the continuously di erentiable and monotone operator with the potential Φ that is the functional de ned on X (its Frechet derivative is the operator F). Moreover, for any x ∈ X the following inequalities hold for any (x, y) , (x , y ) ∈ H × H, z ∈ H and consequently for any (x, y) ∈ H × H the inequation At the beginning for the investigation of the posed problem we set the following expression in order to obtain of the a priori estimations where element y is de ned as the solution of the equation for any t ∈ ( , T) as was already mentioned above. Hence follow ⟨By tt , By t ⟩ + ⟨F (x) , x t ⟩ = ⟨g (x, By t ) , y t ⟩ , where Φ (x) is the functional de ned as Φ (x) = ∫ ⟨F (sx) , x⟩ ds (see, [6]).
Then using condition (iii) on g (x, By t ) in (6) with the initial conditions where C j ≥ are constants independent of x (t). From here follows

By t H (t) + Φ (x (t)) ≤ e tC By
This gives the following estimations for every T ∈ ( , ∞) for a. e. t ∈ ( , T), i.e. y = A − x is contained in the bounded subset of the space y ∈ C ( , T; V) ∩ C ( , T; Y), consequently, we obtain that if the weak solution x (t) exists then it belongs to a bounded subset of the space C ( , T; X) ∩ C , T; V * . Hence one can see, that the following inclusion holds by virtue of (5) in the assumption that x = Ay is a solution of the posed problem in the sense of De nition 2.1.

Proof of Theorem 2.2.
In order to prove of the solvability theorem we will use the Faedo-Galerkin approach.
Let the system y k ∞ k= ⊂ Y be total in Y such that it is complete in the spaces Y , V, and also in the spaces X, H. We will seek out of the approximative solutions y m (t), and consequently x m (t), in the form Thus we obtain the following problem for any z ∈ Y and m = , , .... Consequently, with use of the known procedure ([7 -9]) we obtain, y mt ∈ C ( , T; V), y m ∈ C ( , T; Y) and x m ∈ C ( , T; X), x mt ∈ C , T; V * , moreover, they are contained in the bounded subset of these spaces for any m = , , .... Hence from (9) we get Thus we obtain, that the sequence {x m } ∞ m= of the approximated solutions of the problem is contained in a bounded subset of the space or {x m } ∞ m= such that for a. e. t ∈ ( , T) the following inclusions take place , By mt (t))} ∞ m= have weakly converging subsequences to η (t) and θ (t) in X * and H, respectively, for a. e. t ∈ ( , T). Hence one can pass to the limit in (11) with respect to m ↗ ∞. Then we obtain the following equation It remained to show the following: if the sequence {x m (t)} ∞ m= ≡ {Ay m (t)} ∞ m= is weakly converging to x (t) = Ay (t) then η (t) = F (x (t)) and θ (t) = g (x (t) , By t (t)). In order to show these equations are ful lled we will use the monotonicity of F and the condition (iii).

Conclusion
In this article, the existence of a very weak solution for di erential-operator equations of second order with nonlinear operator in the main part is proved. We would like to note that, in particular, if A is the di erential operator this equation becomes a hyperbolic equation. Consequently, one can investigate previously not studied nonlinear hyperbolic equations with the use of results and the approach presented in this article.
The following work will be focused on nonlinear hyperbolic equations with the nonlinearity of the same type as studied here. Moreover, here the long-time behavior of the very weak solution of the problem is proved, and also the dependence of the behavior of the solution from initial datums is shown. In other words, here we show the behavior of the weak semi-ow (in some sense), de ned by the considered problem, with respect to t when t → +∞.