Classical Solution of the First Mixed Problem for the Telegraph Equation with a Nonlinear Potential

Korzyuk, V. I.; Rudzko, J. V.

doi:10.1134/s0012266122020045

Cited by 9 publications

(14 citation statements)

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“…Repeating the reasoning of the paper [32], we derive conditions (3.2) -(3.4) based on equalities (3.6) and representations (3.1).…”

Section: Integro-differential Equationsmentioning

confidence: 93%

“…Compatibility conditions obtained in this approach are obviously not necessary but sufficient. On the other hand, there are many works [32,33,34,35,36,37,38], where theorems guaranteeing the existence and uniqueness of classical solutions with minimal and unimprovable (i.e., necessary and sufficient) compatibility conditions, are obtained by combining the method of characteristics and fixed point principles (usually the Leray-Schauder fixed point theorem or the Banach fixed point theorem and its generalizations).…”

Section: Modern State Of the Problemmentioning

confidence: 99%

“…• f (t, x, u, u t , u x ) = g(t, x, u), where the function g satisfies a condition of the Lipschitz-Carathéodory type in the third variable with a function k ∈ L 2 loc , i.e., the inequality |g(t, x, u) − g(t, x, z)| ≤ k(t, x)|u − z| holds [32,33,34] (classical solution in the space C 2 ).…”

Section: Modern State Of the Problemmentioning

confidence: 99%

“…• f (t, x, u, u t , u x ) = g(t, x, u) + bu t , where the function g satisfies a condition of the Lipschitz-Carathéodory type in the third variable with a function k ∈ L 2 loc [35] (classical solution in the space C 2 and mild solution in the space C).…”

Section: Modern State Of the Problemmentioning

confidence: 99%

“…We can consider the problem (1.1) -(1.4) as two connected initial boundary value problems: the first mixed problem and the second mixed problem. Then, in some cases, it is possible to obtain a classical [33,35] or generalized [41] solution to the problem (1.1) -(1.4) within one method, but it may have some difficulties if the compatibility conditions are not met. Therefore, in this work, we will consider the problem (1.1) -(1.4) as a single whole.…”

Section: Modern State Of the Problemmentioning

confidence: 99%

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Classical Solutions of a Mixed Problem with the Zaremba Boundary Condition for a Mildly Quasilinear Wave Equation

Korzyuk,

Rudzko

2024

Preprint

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For a mildly nonlinear wave equation given in the first quadrant, we consider a mixed problem in which we pose the Cauchy conditions on the spatial half-line and the Zaremba condition on the time half-line. Using the method of characteristics, we construct the solution in an implicit analytical form as a solution of some integro-differential equations. The solvability of these equations is studied, as well as the dependence on the initial data and the smoothness of their solutions. For the problem in question, the uniqueness of the solution is proved, and the conditions under which its classical solution exists are established. If the compatibility conditions are not satisfied, then we consider a problem with conjugation conditions. We construct a mild solution if the data are not smooth enough. We apply the obtained mathematical results to solve a problem from combustion theory. Mathematics Subject Classification (2010). Primary 35L20; Secondary 35A02, 35A09, 35D99, 35L71, 35Q79, 80A25.

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“…Repeating the reasoning of the paper [32], we derive conditions (3.2) -(3.4) based on equalities (3.6) and representations (3.1).…”

Section: Integro-differential Equationsmentioning

confidence: 93%