We consider a reaction-diffusion system of the activator-inhibitor type with unilateral boundary conditions leading to a quasivariational inequality. We show that there exists a positive eigenvalue of the problem and we obtain an instability of the trivial solution also in some area of parameters where the trivial solution of the same system with Dirichlet and Neumann boundary conditions is stable. Theorems are proved using the method of a jump in the Leray-Schauder degree.

This paper investigates the inverse problem of finding the time-dependent diffusion coefficient in a quasilinear parabolic equation with the nonlocal boundary and integral overdetermination conditions. Under some natural regularity and consistency conditions on the input data the existence, uniqueness and continuously dependence upon the data of the solution are shown. Finally, some numerical experiments are presented.

The problem of determining the source term $F(x,t)$ in the linear parabolic equation $u_t={\left(k\left(x\right)u_x(x,t)\right)}_x+F(x,t)$ from the measured data at the final time $u(x,T)=\mu \left(x\right)$ is formulated. It is proved that the Fréchet derivative of the cost functional $J\left(F\right)=\parallel {\mu }_T{\left(x\right)-u(x,T)\parallel }_0^2$ can be formulated via the solution of the adjoint parabolic problem. Lipschitz continuity of the gradient is proved. An existence result for a quasi solution of the considered inverse problem is proved. A monotone iteration scheme is obtained based on the gradient method. Convergence rate is proved....

Nonlinear parabolic functional differential equations with initial boundary conditions of the Neumann type are considered. A general class of difference methods for the problem is constructed. Theorems on the convergence of difference schemes and error estimates of approximate solutions are presented. The proof of the stability of the difference functional problem is based on a comparison technique. Nonlinear estimates of the Perron type with respect to the functional variable for given functions...

Let $\Omega$ be a bounded open subset of ${ℝ}^n$, let $X=(x,t)$ be a point of ${ℝ}^n\times {ℝ}^N$. In the cylinder $Q=\Omega \times (-T,0)$, $T>0$, we deduce the local differentiability result $$u\in L^2(-a,0,H^2(B\left(\sigma \right),{ℝ}^N))\cap H^1(-a,0,L^2(B\left(\sigma \right),{ℝ}^N))$$ for the solutions $u$ of the class $L^q(-T,0,H^{1,q}(\Omega ,{ℝ}^N))\cap C^{0,\lambda }(\overline{Q},{ℝ}^N)$ ($0<\lambda <1$, $N$ integer $\ge 1$) of the nonlinear parabolic system $$-\sum _{i=1}^n{D}_i{a}^i(X,u,Du)+{\displaystyle \frac{\partial u}{\partial t}}=B^0(X,u,Du)$$ with quadratic growth and nonlinearity $q\ge 2$. This result had been obtained making use of the interpolation theory and an imbedding theorem of Gagliardo-Nirenberg type for functions $u$ belonging to $W^{1,q}\cap C^{0,\lambda }$.