Criterion for a generalized solution in the class $L_p$ for the wave equation to be in the class $W^l_p$

I. S. Mokrousov

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 18 (2018) no. 3, pp. 297-304

Résumé

In this paper we consider the question of whether a generalized solution of the wave equation belongs to different function spaces. Consideration of classical solutions imposes substantial restrictions on the initial data of the problem. But if we proceed not from differential but from integral equations, then the class of solutions and the class of initial boundary value problems can be substantially expanded. To solve the boundary value problem for the wave equation obtained by the wave counting method, it is easy to obtain a sufficient condition for belonging to a particular class. A much more subtle question is finding the necessary and sufficient condition. A criterion is established for class $W_p^l$ to belong to the solution of the wave equation generalized from class $L_p$. The criterion establishes a connection between the condition for the boundary function $\mu(t)$ and the condition on the solution of the problem $ u_{tt}(x, t) - u_{xx}(x, t) = 0 $. Thus, this criterion can be applied to the estimation of control tasks, in particular, depending on the final condition of the problem, it is possible to establish the membership of the control function. In the same way this criterion can be applied to the estimates of the observation problems for the wave equation, when the properties of the solution can be predicted from the properties of the boundary function. This article consists of the formulation of the problem, the consideration of earlier results, the formulation and proof of the main theorem. The proof of the main theorem is essentially based on the representation of the solution of this problem in explicit analytical form. This result generalizes the previously obtained criterion for $ W_p^1 $. It should be noted that, although the proof of the criterion for the class $ W_p ^ l $ structurally repeats the proof for the class $ W_p ^ 1 $, it is significantly complicated by more subtle estimates of the norms of functions entering into the solution of the problem.