Abstract
AbstractWe study a nonlinear evolutionary partial differential equation that can be viewed as a generalization of the heat equation where the temperature gradient is a priori bounded but the heat flux provides merely $$L^1$$
L
1
-coercivity. Applying higher differentiability techniques in space and time, choosing a special weighted norm (equivalent to the Euclidean norm in $$\mathbb {R}^d$$
R
d
), incorporating finer properties of integrable functions and flux truncation techniques, we prove long-time and large-data existence and uniqueness of weak solution, with an $$L^1$$
L
1
-integrable flux, to an initial spatially-periodic problem for all values of a positive model parameter. If this parameter is smaller than $$2/(d+1)$$
2
/
(
d
+
1
)
, where d denotes the spatial dimension, we obtain higher integrability of the flux. As the developed approach is not restricted to a scalar equation, we also present an analogous result for nonlinear parabolic systems in which the nonlinearity, being the gradient of a strictly convex function, gives an a-priori $$L^\infty $$
L
∞
-bound on the gradient of the unknown solution.
Funder
Grantová Agentura Ceské Republiky
Deutsche Forschungsgemeinschaft
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,Analysis
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