On evolutionary problems with a-priori bounded gradients

We 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 -coercivity. Applying higher differentiability techniques in space and time, choosing a special weighted norm (equivalent to the Euclidean norm in 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 -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) , 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^∞ -bound on the gradient of the unknown solution.