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@MISC{,
            author = {Bul{\'{\i}}{\v c}ek, Miroslav and Hru{\v s}ka, David and M{\'{a}}lek, Josef},
          keywords = {$\infty$-Laplacian, a~priori bounded gradient, existence, nonlinear parabolic equation, renormalized solution, uniqueness, weak solution},
             title = {On evolutionary problems with a-priori bounded gradients},
              year = {2021},
          abstract = {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 \mbox{$L^1$-coercivity}. Applying higher differentiability techniques in space and time, choosing a special weighted norm (equivalent to the Euclidean norm in $\mathbb{R}^d$), incorporating finer properties of integrable functions and using the concept of renormalized solution, 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  $L^\infty$-bound on the gradient of the unknown solution.},
  Preprint project = {NCMM},
     Preprint year = {2021},
   Preprint number = {02},
       Preprint ID = {NCMM/2021/02}
}