Global nonexistence and blow-up results for a quasi-linear evolution equation with variable-exponent nonlinearities

"In this paper, we consider a class of quasi-linear parabolic equations with variable exponents, $$a\left( x,t\right) u_{t}-\Delta _{m\left( .\right) }u=f_{p\left( .\right)}\left( u\right)$$ in which $f_{p\left( .\right)}\left( u\right)$ the source term, $a(x,t)>0$ is a nonnegative function, and the exponents of nonlinearity $m(x)$, $p(x)$ are given measurable functions. Under suitable conditions on the given data, a finite-time blow-up result of the solution is shown if the initial datum possesses suitable positive energy, and in this case, we precise estimate for the lifespan $T^{\ast }$ of the solution. A blow-up of the solution with negative initial energy is also established."

Stability results for weak solutions to backward one-dimensional semi-linear parabolic equations with locally Lipschitz source

Stability estimates of Hölder type for weak solutions to backward one-dimensional semi-linear parabolic equations with locally Lipschitz source are obtained. It is noticed that stability results for weak solutions to nonlinear inverse problems are very rare in the literature.

State-constrained controllability of linear reaction-diffusion systems

We study the controllability of a coupled system of linear parabolic equations, with nonnegativity constraint on the state. We establish two results of controllability to trajectories in large time: one for diagonal diffusion matrices with an “approximate” nonnegativity constraint, and a another stronger one, with “exact” nonnegativity constraint, when all the diffusion coefficients are equal. The proofs are based on a “staircase” method. Finally, we show that state-constrained controllability admits a positive minimal time, even with weaker unilateral constraint on the state.

MESHLESS GALERKIN METHOD BASED ON RBFS AND REPRODUCING KERNEL FOR QUASI-LINEAR PARABOLIC EQUATIONS WITH DIRICHLET BOUNDARY CONDITIONS

The main aim of this paper is to present a hybrid scheme of both meshless Galerkin and reproducing kernel Hilbert space methods. The Galerkin meshless method is a powerful tool for solving a large class of multi-dimension problems. Reproducing kernel Hilbert space method is an extremely efficient approach to obtain an analytical solution for ordinary or partial differential equations appeared in vast areas of science and engineering. The error analysis and convergence show that the proposed mixed method is very efficient. Since the solution space spanned by radial basis functions do not directly satisfy essential boundary conditions, an auxiliary parameterized technique is employed. Theoretical studies indicate that this new method is very stable, though a parameterized problem is employed instead of the main problem.

Stability and convergence of second order backward differentiation schemes for parabolic Hamilton–Jacobi–Bellman equations

We study a second order Backward Differentiation Formula (BDF) scheme for the numerical approximation of linear parabolic equations and nonlinear Hamilton–Jacobi–Bellman (HJB) equations. The lack of monotonicity of the BDF scheme prevents the use of well-known convergence results for solutions in the viscosity sense. We first consider one-dimensional uniformly parabolic equations and prove stability with respect to perturbations, in the $$L^2$$ L 2 norm for linear and semi-linear equations, and in the $$H^1$$ H 1 norm for fully nonlinear equations of HJB and Isaacs type. These results are then extended to two-dimensional semi-linear equations and linear equations with possible degeneracy. From these stability results we deduce error estimates in $$L^2$$ L 2 norm for classical solutions to uniformly parabolic semi-linear HJB equations, with an order that depends on their Hölder regularity, while full second order is recovered in the smooth case. Numerical tests for the Eikonal equation and a controlled diffusion equation illustrate the practical accuracy of the scheme in different norms.

Gradient weighted norm inequalities for very weak solutions of linear parabolic equations with BMO coefficients

In this paper, we give a short proof of the Lorentz estimates for gradients of very weak solutions to the linear parabolic equations with the Muckenhoupt class A q -weights u t − div ( A ( x , t ) ∇ u ) = div ( F ) , in a bounded domain Ω × ( 0 , T ) ⊂ R N + 1 , where A has a small mean oscillation, and Ω is a Lipchistz domain with a small Lipschitz constant.

Second order optimality conditions for periodic optimal control problems governed by semilinear parabolic differential equations

In this paper, we study second-order optimality conditions for some optimal control problems governed by some semi-linear parabolic equations with periodic state constraint in time. We obtain a necessary condition and a sufficient condition in terms of the second order derivative of the associated Lagrangian. These two conditions correspond to the positive definite and the nonnegativity of the second order derivative of the Lagrangian on the same cone, respectively.