We develop a new method for proving the existence of a boundary trace, in the class of Borel measures, of nonnegative solutions of $-\Delta u+g(x,u)\ge 0$ in a smooth domain $\Omega$ under very general assumptions on $g$. This new definition which extends the previous notions of boundary trace is based upon a sweeping technique by solutions of Dirichlet problems with measure boundary data. We also prove a boundary pointwise blow-up estimate of any solution of such inequalities in terms of the Poisson kernel. If the nonlinearity...

We consider a one-dimensional semilinear parabolic equation with a gradient nonlinearity. We provide a complete classification of large time behavior of the classical solutions $u$: either the space derivative $u_x$ blows up in finite time (with $u$ itself remaining bounded), or $u$ is global and converges in $C^1$ norm to the unique steady state. The main difficulty is to prove $C^1$ boundedness of all global solutions. To do so, we explicitly compute a nontrivial Lyapunov functional by carrying out the method of...

This paper deals with parabolic-elliptic chemotaxis systems with the sensitivity function $\chi \left(v\right)$ and the growth term $f\left(u\right)$ under homogeneous Neumann boundary conditions in a smooth bounded domain. Here it is assumed that $0<\chi \left(v\right)\le {\chi }_0/v^k$$(k\ge 1...$

We prove local in time solvability of the nonlinear initial-boundary problem to nonlinear nondiagonal parabolic systems of equations (multidimensional case). No growth restrictions are assumed on generating the system functions. In the case of two spatial variables we construct the global in time solution to the Cauchy-Neumann problem for a class of nondiagonal parabolic systems. The solution is smooth almost everywhere and has an at most finite number of singular points.

The existence, uniqueness and regularities of the generalized global solutions and classical global solutions to the equation $$u_t=-A\left(t\right)u_{x^4}+B\left(t\right)u_{x^2}+g{\left(u\right)}_{x^2}+f{\left(u\right)}_x+h{\left(u_x\right)}_x+G\left(u\right)$$ with the initial boundary value conditions $$u(-\ell ,t)=u(\ell ,t)=0,u_{x^2}(-\ell ,t)=u_{x^2}(\ell ,t)=0,u(x,0)=\varphi \left(x\right),$$ or with the initial boundary value conditions $$u_x(-\ell ,t)=u_x(\ell ,t)=0,u_{x^3}(-\ell ,t)=u_{x^3}(\ell ,t)=0,u(x,0)=\varphi \left(x\right),$$ are proved. Moreover, the asymptotic behavior of these solutions is considered under some conditions.

In this paper, we obtain some existence theorems of nonnegative solutions with compact support for homogeneous Dirichlet elliptic problems; we also extend these results to parabolis systems.Supersolution and comparison principles are our main ingredients.

A nonlinear parabolic problem with the Newton boundary conditions and its weak formulation are examined. The problem describes nonstationary heat conduction in inhomogeneous and anisotropic media. We prove a comparison principle which guarantees that for greater data we obtain, in general, greater weak solutions. A new strategy of proving the comparison principle is presented.

A class of quasilinear parabolic systems with quadratic nonlinearities in the gradient is considered. It is assumed that the elliptic operator of a system has variational structure. In the multidimensional case, the behavior of solutions of the Cauchy-Dirichlet problem smooth on a time interval $[0,T)$ is studied. Smooth extendibility of the solution up to $t=T$ is proved, provided that “normilized local energies” of the solution are uniformly bounded on $[0,T)$. For the case where $[0,T)$ determines the maximal interval...

We present here a discretization of a nonlinear oblique derivative boundary value problem for the heat equation in dimension two. This finite difference scheme takes advantages of the structure of the boundary condition, which can be reinterpreted as a Burgers equation in the space variables. This enables to obtain an energy estimate and to prove the convergence of the scheme. We also provide some numerical simulations of this problem and a numerical study of the stability of the scheme, which appears...

We present here a discretization of a nonlinear oblique derivative boundary value problem for the heat equation in dimension two. This finite difference scheme takes advantages of the structure of the boundary condition, which can be reinterpreted as a Burgers equation in the space variables. This enables to obtain an energy estimate and to prove the convergence of the scheme. We also provide some numerical simulations of this problem and a numerical study of the stability of the scheme, which appears...