Interior and boundary higher integrability of very weak solutions for quasilinear parabolic equations with variable exponents

Adimurthi K, Byun S-S, Oh J (2020)
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS 194: 111370.

Zeitschriftenaufsatz | Veröffentlicht | Englisch
 
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Autor*in
Adimurthi, Karthik; Byun, Sun-Sig; Oh, JehanUniBi
Abstract / Bemerkung
We prove boundary higher integrability for the (spatial) gradient of very weak solutions of quasilinear parabolic equations of the form {u(t) - divA(x, t,del u) = 0 on Omega x (-T, T), u = 0 on partial derivative Omega x (-T, T), where the non-linear structure A(x,t, del u) is modeled after the variable exponent p(x, t)-Laplace operator given by vertical bar del u vertical bar(p(x, t)-2)del u. To this end, we prove that the gradients satisfy a reverse Holder inequality near the boundary by constructing a suitable test function which is Lipschitz continuous and preserves the boundary values. In the interior case, such a result was proved in Verena Bogelein and Qifan Li (2014) provided p(x, t) >= p(-) >= 2 holds and was then extended to the singular case 2n/n+2 < p(-) <= p(x, t) <= p(+)<= 2 in Qifan Li (2017). This restriction was necessary because the intrinsic scalings for quasilinear parabolic problems are different in the case p(+) <= 2 andp(-) >= 2. In this paper, we develop a new approach, using which, we are able to extend the results of Verena Bogelein and Qifan Li (2014), Qifan Li (2017) to the full range 2n/n+2 < p(-) <= p(x, t)<= p(+)
Stichworte
Quasilinear parabolic equations; Unified intrinsic scaling; Boundary; higher integrability; Very weak solutions; p(x; t)-Laplacian; Variable; exponent spaces
Erscheinungsjahr
2020
Zeitschriftentitel
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS
Band
194
Art.-Nr.
111370
ISSN
0362-546X
eISSN
1873-5215
Page URI
https://pub.uni-bielefeld.de/record/2942070

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Adimurthi K, Byun S-S, Oh J. Interior and boundary higher integrability of very weak solutions for quasilinear parabolic equations with variable exponents. NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS. 2020;194: 111370.
Adimurthi, K., Byun, S. - S., & Oh, J. (2020). Interior and boundary higher integrability of very weak solutions for quasilinear parabolic equations with variable exponents. NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 194, 111370. doi:10.1016/j.na.2018.10.014
Adimurthi, K., Byun, S. - S., and Oh, J. (2020). Interior and boundary higher integrability of very weak solutions for quasilinear parabolic equations with variable exponents. NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS 194:111370.
Adimurthi, K., Byun, S.-S., & Oh, J., 2020. Interior and boundary higher integrability of very weak solutions for quasilinear parabolic equations with variable exponents. NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 194: 111370.
K. Adimurthi, S.-S. Byun, and J. Oh, “Interior and boundary higher integrability of very weak solutions for quasilinear parabolic equations with variable exponents”, NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, vol. 194, 2020, : 111370.
Adimurthi, K., Byun, S.-S., Oh, J.: Interior and boundary higher integrability of very weak solutions for quasilinear parabolic equations with variable exponents. NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS. 194, : 111370 (2020).
Adimurthi, Karthik, Byun, Sun-Sig, and Oh, Jehan. “Interior and boundary higher integrability of very weak solutions for quasilinear parabolic equations with variable exponents”. NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS 194 (2020): 111370.