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Existence of variational solutions to doubly nonlinear systems in general noncylindrical domains
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A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä
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en
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57
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Journal of Differential Equations, Volume 462, pp. 1-57
Abstract
We consider the Cauchy–Dirichlet problem to doubly nonlinear systems of the form ∂t(|u|q−1u)−div(Dξf(x,u,Du))=−Duf(x,u,Du) with q∈(0,∞) in a bounded noncylindrical domain E⊂Rn+1. Further, we suppose that x↦f(x,u,ξ) is integrable, that (u,ξ)↦f(x,u,ξ) is convex, and that f satisfies a p -growth and -coercivity condition for some p>max{1,n(q+1)n+q+1}. Merely assuming that Ln+1(∂E)=0, we prove the existence of variational solutions u∈L∞(0,T;Lq+1(E,RN)). If E does not shrink too fast, we show that for the solution u constructed in the first step, |u|q−1u admits a distributional time derivative. Moreover, under suitable conditions on E and the stricter lower bound p≥(n+1)(q+1)n+q+1, u is continuous with respect to time.
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Publisher Copyright: © 2026 The Author(s).
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Schätzler, L, Scheven, C, Siltakoski, J & Stanko, C 2026, 'Existence of variational solutions to doubly nonlinear systems in general noncylindrical domains', Journal of Differential Equations, vol. 462, 114139, pp. 1-57. https://doi.org/10.1016/j.jde.2026.114139