On evolutionary problems with a-priori bounded gradients
نویسندگان
چکیده
Abstract We study a nonlinear evolutionary partial differential equation that can be viewed as generalization of the heat where temperature gradient is priori bounded but flux provides merely $$L^1$$ L 1 -coercivity. Applying higher differentiability techniques in space and time, choosing special weighted norm (equivalent to Euclidean $$\mathbb {R}^d$$ R d ), incorporating finer properties integrable functions truncation techniques, we prove long-time large-data existence uniqueness weak solution, with an -integrable flux, initial spatially-periodic problem for all values positive model parameter. If this parameter smaller than $$2/(d+1)$$ 2 / ( + ) , d denotes spatial dimension, obtain integrability flux. As developed approach not restricted scalar equation, also present analogous result parabolic systems which nonlinearity, being strictly convex function, gives a-priori $$L^\infty $$ ∞ -bound on unknown solution.
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ژورنال
عنوان ژورنال: Calculus of Variations and Partial Differential Equations
سال: 2023
ISSN: ['0944-2669', '1432-0835']
DOI: https://doi.org/10.1007/s00526-023-02524-4