We consider a quasilinear parabolic differential equation associated with the renormalization group transformation of the two–dimensional hierarchical Coulomb system in the limit as the size of the block L ↓ 1. We show that the initial value problem is well defined in a suitable function space and the solution converges, as t → ∞, to one of the countably infinite equilibrium solutions. The j–th...

This article concerns the asymptotic behavior of solutions to the Cauchy problem of a degenerate quasilinear parabolic equations with a gradient term. A blow-up theorem of Fujita type is established and the critical Fujita exponent is formulated by the spacial dimension and the behavior of the coefficient of the gradient term at ∞.

We study a two-point free boundary problem for a quasilinear parabolic equation. This problem arises in the model of flame propagation in combustion theory. It also arises in the study of the motion of interface moving with curvature in which the studied problem is confined in the conical region bounded by two straight lines and the interface has prescribed touching angles with these two straig...

This paper develops and analyzes a semi-discrete and a fully discrete finite element method for a one-dimensional quasilinear parabolic stochastic partial differential equation (SPDE) which describes the stochastic mean curvature flow for planar curves of graphs. To circumvent the difficulty caused by the low spatial regularity of the SPDE solution, a regularization procedure is first proposed ...

Using the abstract framework of [10] we analyze a residual a posteriori error estimator for space-time finite element discretizations of parabolic pdes. The estimator gives global upper and local lower bounds on the error of the numerical solution. The finite element discretizations in particular cover the so-called θ-scheme, which includes the implicit and explicit Euler methods and the Crank-...

‎we study the existence of soliton solutions for a class of‎ ‎quasilinear elliptic equation in $mathbb{textbf{r}}^2$ with critical exponential growth‎. ‎this model has been proposed in the self-channeling of a‎ ‎high-power ultra short laser in matter‎.

in a pointwise sense and in a viscosity sense. Here uν denotes the derivative of u with respect to the inward unit spatial normal ν to the free boundary ∂{u > 0}, M = ∫ β(s) ds, α(ν,M) := Φ−1 ν (M) andΦν(α) := −A(αν)+αν ·F(αν), where A(p) is such that F(p) = ∇A(p) with A(0) = 0. Some of the results obtained are new even when the operator under consideration is linear. 2000 Mathematics Subject C...

We consider a quasilinear parabolic problem u′(t) = Q ( u(t) ) u(t), u(t0) = u0 ∈ D, where Q(w) : D ⊂ X → X, w ∈ W ⊂ X, is a family of sectorial operators in a Banach space X with fixed domain D. This problem is discretized in time by means of a strongly A(θ)-stable, 0 < θ ≤ π/2, Runge–Kutta method. We prove that the resulting discretization is stable, under some natural assumptions on the depe...

This paper is concerned with the state-constrained optimal control of the threedimensional thermistor problem, a fully quasilinear coupled system of a parabolic and elliptic PDE with mixed boundary conditions. This system models the heating of a conducting material by means of direct current. Local existence, uniqueness and continuity for the state system are derived by employing maximal parabo...