Sewing method for weak solutions of second-order hyperbolic equation with variable domains of discontinuous unbounded operators

• Published in: Differential equations Vol. 50; no. 5; pp. 643 - 654
• Main Author: Lomovtsev, F. E.
• Format: Journal Article
• Language: English
• Published: Moscow Pleiades Publishing 01-05-2014; Springer Nature B.V
• Subjects: Banach spaces; Cauchy problem; Cauchy problems; Derivatives; Difference and Functional Equations; Differential equations; Discontinuity; Intervals; Matching; Mathematical analysis; Mathematical functions; Mathematical models; Mathematics; Mathematics and Statistics; Operators; Ordinary Differential Equations; Partial Differential Equations; Sewing; Studies; Theorems; Cauchy Problem; Global Weak Solution; Weak Solution; Variable Domain; Energy Inequality
• ISSN: 0012-2661; 1608-3083
• DOI: 10.1134/S0012266114050073

We prove the existence and uniqueness of global weak solutions on the entire interval for the Cauchy problem for hyperbolic differential-operator equations with time-discontinuous operators that have variable domains and satisfy certain matching conditions at the points of discontinuity. To this end, we develop a method of successive sewing of existing local weak solutions of Cauchy problems on the smoothness intervals of the operators. The sewing method is based on special energy inequalities, which imply the time continuity of local weak solutions in the main space and of their first derivatives in some negative spaces and hence the existence of the corresponding limit values at the points of discontinuity. These values, with regard for the matching conditions, are taken for the initial data on each successive interval.