Weak solution for nonlocal thermistor problem in generalized Sobolev spaces
Keywords:
Thermistor problem, topological degree, weak solution, generalized Sobolev spacesAbstract
We establish by using topological degree method in the framework of generalized Sobolev spaces the existence of at least weak solution for nonlocal Dirichlet thermistor problem associated to the equation
\dfrac{\partial u}{\partial t} -\text{div}\big(a(x,t,u,\nabla
u)\big)=\displaystyle\lambda\dfrac{f(u)}{\big(\int_\Omega
f(s)\,ds\big)^2}
where $-\text{div}\big(a(x,t,u,\nabla u)\big)$ is a divergence operator of Leray-Lions type defined from the energy space $\mathcal{H}\subset L^{p^-}(0,T,W^{1,p(\cdot)}(\Omega))$ into its dual space and $f>0$.
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[1] M. Ait Hammou and E. Azroul. Strongly nonlinear elliptic problem by topological degree in generalized sobolev spaces. vNonlinear Studies (MESA), 12(1), 2021. 4
[2] S. I. Araz. A fractional optimal control problem with final observation governed by wave equation. Chaos, 30(3-4): 251–258, 2019. 4
[3] S. I. Araz and M. Suba¸si. On the control of coefficient function in a hyperbolic problem with dirichlet conditions. International Journal of Differential Equations, 7(1):74–77, 2018. 4
[4] E. Azroul, B. Lahmi, and K. El Haiti. Nonlinear degenerated elliptic problems with dual data and nonstandard growth. Mathematical reports, 20(70):81–91, 2018. 3.2, 3.2
[5] M. Bendahmane, P. Wittbold, and A. Zimmermann. Renormalized solutions for a nonlinear parabolic equation with variable exponents and L1-data. Journal of Differential Equations, 249(6):1483–1515, 2010. 2.1
[6] J. Berkovits. Extension of the leray-schauder degree for abstract hammerstein type mappings. Journal of Differential Equations, 234:289–310, 2007. 2.3, 2.2
[7] I. Dahi and M. R. Sidi Ammi. Existence of a weak solution for nonlocal thermistor problem in sobolev spaces with variable exponent. Filomat, 39(11):3779–3787, 2025. 1, 3.2
[8] A. El Hachimi and M. R. Sidi Ammi. Thermistor problem: a nonlocal parabolic problem. Electronic Journal of Differential Equations [electronic only], 2004:117–128, 2004. 1, 1
[9] X. Fan and D. Zhao. On the spaces Lp(x)(Ω) and Wm,p(x)(Ω). Journal of Mathematical Analysis and Applications, 263(2):424–446, 2001. 2.1, 2.1, 2.1
[10] M. T. Gonz´alez Montesinos and F. Orteg´on Gallego. Existence of a capacity solution to a coupled nonlinear parabolic elliptic system. Communications on Pure and Applied Analysis, 6(1):23–42, 2007. 1, 1, 4
[11] O. Kov`aˇcik and J. R´akosnik. On spaces Lp(x) and Wk,p(x). Czechoslovak Mathematical Journal, 41(4):592–618, 1991. 2.1,2.1, 2.1
[12] A. A. Lacey. Diffusion models with blow-up. Journal of Computational and Applied Mathematics, 97:39–49, 1998. 1
[13] M. Suba¸si and S. I. Araz. Numerical regularization of optimal control for the coefficient function in a wave equation. Iranian Journal of Science and Technology, Transactions A: Science, 43:2325–2333, 2019. 4
[14] X. Xu. A strongly degenerate system involving an equation of parabolic type and an equation of elliptic type. Communications in Partial Differential Equations, 18:199–213, 1993. 4
[15] Z. Zeidler. Nonlinear functional analysis and its applications, ii/b: Nonlinear monotone operators. In Springer, New York, 1990. 3.2
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Copyright (c) 2026 Ouejdane Baali, Manar Chahboune, Badr Lahmi, Mohamed Zitane
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