| Abstract: |
| This study investigates the well-posedness of solutions to the initial boundary value problem for parabolic equations with variable exponents of multiple regime (subcritical, critical, and supercritical) on an annulus. The presence of critical and supercritical regimes disrupts classical Sobolev embeddings, leading to the lack of compactness. To address these issues, we use the Strauss inequality to restore compact Sobolev embeddings for radially symmetric functions. By employing the subdifferential technique with symmetry constraints, we establish local existence of solutions for any radially symmetric initial data and demonstrate uniqueness. We pioneer the application of the potential well theory to classify initial data based on three energy levels: subcritical, critical, and supercritical. For subcritical and critical levels, we analyze cases with non-positive and positive initial energy, obtaining results on finite-time blowup and identifying threshold conditions for global existence versus blowup. Finally, we extend these results to a broader class of locally symmetric domains containing an annulus. |
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