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Global Well-Posedness of Solutions to a Class of Double Phase Parabolic Equation With Variable Exponents

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Abstract

The main objective of this paper is to study a class of parabolic equation driven by double phase operator with initial-boundary value conditions. As is well known, subcritical hypotheses play an important role in investigating well-posedness result to parabolic and elliptic equations. The highlight of this paper is to overcome the difficulties without subcritical assumption creates by restricting the domain. We firstly obtain the local solution separately on the radial and the nonradial cases by the appropriate approach of subdifferential, Palais principle of symmetric criticality and variational methods. Later, using the potential well method, the results of global solution and decay estimates of energy functional are proved when the initial energy is subcritical. Finally, we derived the blow-up in finite time of solutions when the initial data satisfies different conditions. The present work extends and complements some of earlier contributions related parabolic equations involving p(x)-Laplacian.

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Acknowledgements

This research is supported by the National Natural Science Foundation of China (No. 11201095) and the Fundamental Research Funds for the Central Universities (No. 3072022TS2402).

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  1. School of Mathematical Sciences, Harbin Engineering University, Harbin, 150001, P.R. China

    Wen-Shuo Yuan, Bin Ge & Qing-Hai Cao

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  1. Wen-Shuo Yuan
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  2. Bin Ge
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  3. Qing-Hai Cao
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Correspondence to Bin Ge.

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Yuan, WS., Ge, B. & Cao, QH. Global Well-Posedness of Solutions to a Class of Double Phase Parabolic Equation With Variable Exponents. Potential Anal 60, 1007–1030 (2024). https://doi.org/10.1007/s11118-023-10077-6

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