Around a singular solution of a nonlocal nonlinear heat equation

Biler, Piotr; Pilarczyk, Dominika

doi:10.1007/s00030-019-0552-z

Cited by 4 publications

(2 citation statements)

References 41 publications

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“…Definition 2.5. Following [43,44,45], we call a function u representable in the form (2.7), (2.8), (2.14), (2.10), where γ : [0, ∞) ∋ t → γ(t; ϕ(0) − µ 1 (0)) ∈ R is a function satisfying the condition (2.13), a mild solution of the problem (1.1) -(1.4).…”

Section: Resultsmentioning

confidence: 99%

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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Section: Resultsmentioning

confidence: 99%

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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show abstract

“…α , implies that problem ( 5)-( 6 with the constant and/or suitable pointwise estimates comparing the initial condition u 0 with the singular solution u ∞ , see e.g. [7]. One of the results in this direction is [7, Theorem 2.6]: if…”

Section: Corollary 4 (Dichotomy)mentioning

confidence: 99%