We study a higher order parabolic partial differential equation that arises in the context of condensed matter physics. It is a fourth order semilinear equation which nonlinearity is the determinant of the Hessian matrix of the solution. We consider this model in a bounded domain of the real plane and study its stationary solutions both when the geometry of this domain is arbitrary and when it is the unit ball and the solution is radially symmetric. We also consider the initial-boundary value problem for the full parabolic equation. We summarize our results on existence of solutions in these cases and propose an open problem related to the existence of self-similar solutions.
On the segment $I=[a,b]$ consider the problem \[ u^{\prime }(t)=f(u)(t) , \quad u(a)=c, \] where $f\:C(I,\mathbb{R})\rightarrow L(I,\mathbb{R})$ is a continuous, in general nonlinear operator satisfying Carathéodory condition, and $c\in \mathbb{R}$. The effective sufficient conditions guaranteeing the solvability and unique solvability of the considered problem are established. Examples verifying the optimality of obtained results are given, as well.
Consider the homogeneous equation $$ u'(t)=\ell (u)(t)\qquad \mbox {for a.e. } t\in [a,b] $$ where $\ell \colon C([a,b];\Bbb R)\to L([a,b];\Bbb R)$ is a linear bounded operator. The efficient conditions guaranteeing that the solution set to the equation considered is one-dimensional, generated by a positive monotone function, are established. The results obtained are applied to get new efficient conditions sufficient for the solvability of a class of boundary value problems for first order linear functional differential equations.
Nonimprovable, in a sense sufficient conditions guaranteeing the unique solvability of the problem \[ u^{\prime }(t)=\ell (u)(t)+q(t), \qquad u(a)=c, \] where $\ell \:C(I,\mathbb R)\rightarrow L(I,\mathbb R)$ is a linear bounded operator, $q\in L(I,\mathbb R)$, and $c\in \mathbb R$, are established.