Quasilinear Parabolic Equations and Systems with Two Independent Variables

On Quasilinear Non-Uniformly Parabolic Equations in General Form

“…for (t, x) # Q T , |u| <M and any p implies the a priori estimate |u x 1 | <C 1 . In order to obtain the a priori estimate |u x 2 | <C 2 it is sufficient to require that [9,10] (see also [4,11]).…”

“…In the case of one space variable this estimate have been obtained without any restriction on the smoothness of the coefficients (see [1]). In the higher dimensional case the main idea, that goes back to S. N. Bernstein, involves the preliminary boundary estimate of |{u|, differentiation of the equation and application of the maximum principle (see [2]).…”

“…The estimate is independent of the ellipticity coefficient, of the smoothness of the coefficients of the equation and is found explicitly. Based on this a priori estimate and known results [1,3,4] the existence theorem is proved. In deriving an a priori estimate for |{u| the idea of introducing an additional spatial variable [1] is used (see also [5 8]).…”

On the generalized Burgers equation

“…Recall the following estimate (see [4] or [15]): for any classical solution of problem (1.1)-(1.3) the inequality…”

Analysis of a Semilinear PDE for Modeling Static Solutions of Josephson Junctions

“…Note also that the coefficients of the operator are nonsmooth. Using the additional variable method proposed in [11] and [17,18], we first obtain a priori estimates on the gradient of the solution that are of physical interest. We then prove, under certain assumptions, the existence and uniqueness of the solution and the convergence of a fixed point linearization method.…”