We consider the quasilinear equation , and present the proof of the local existence of positive radial solutions near under suitable conditions on . Moreover, we provide a priori estimates of positive radial solutions near when for is bounded near .
We review some recent results concerning a priori bounds for solutions of superlinear parabolic problems and their applications.
We examine an elliptic optimal control problem with control and state constraints in ℝ3. An improved error estimate of 𝒪(hs) with 3/4 ≤ s ≤ 1 − ε is proven for a discretisation involving piecewise constant functions for the control and piecewise linear for the state. The derived order of convergence is illustrated by a numerical example.
We examine an elliptic optimal control problem with control and state constraints in ℝ3. An improved error estimate of 𝒪(hs) with 3/4 ≤ s ≤ 1 − ε is proven for a discretisation involving piecewise constant functions for the control and piecewise linear for the state. The derived order of convergence is illustrated by a numerical example.
We examine an elliptic optimal control problem with control and state constraints in ℝ3. An improved error estimate of 𝒪(hs) with 3/4 ≤ s ≤ 1 − ε is proven for a discretisation involving piecewise constant functions for the control and piecewise linear for the state. The derived order of convergence is illustrated by a numerical example.
We prove a priori estimates for solutions of the Poisson and heat equations in weighted spaces of Kondrat'ev type. The weight is a power of the distance from a distinguished axis.
In this survey we consider superlinear parabolic problems which possess both blowing-up and global solutions and we study a priori estimates of global solutions.