Quasilinear Heat Equation Research Articles

An approximation result for a quasi-linear stochastic heat equation

We study the limiting behavior, as n goes to ∞ , of a solution of a stochastic partial differential equation driven by a process X n which converges in law to the Brownian sheet. Under some assumptions, we prove that the solution u n converges in distribution in C ( [ 0 , 1 ] 2 ) to a weak solution of a SPDE.

Weak Convergence for the Stochastic Heat Equation Driven by Gaussian White Noise

In this paper, we consider a quasi-linear stochastic heat equation with spatial dimension one, with Dirichlet boundary conditions and controlled by the space-time white noise. We formally replace the random perturbation by a family of noisy inputs depending on a parameter that approximate the white noise in some sense. Then, we provide sufficient conditions ensuring that the real-valued mild solution of the SPDE perturbed by this family of noises converges in law, in the space of continuous functions, to the solution of the white noise driven SPDE. Making use of a suitable continuous functional of the stochastic convolution term, we show that it suffices to tackle the linear problem. For this, we prove that the corresponding family of laws is tight and we identify the limit law by showing the convergence of the finite dimensional distributions. We have also considered two particular families of noises to that our result applies. The first one involves a Poisson process in the plane and has been motivated by a one-dimensional result of Stroock. The second one is constructed in terms of the kernels associated to the extension of Donsker's theorem to the plane.

Modeling general laws of spatial–temporal evolution of society: hyperbolic growth and historical cycles

The article investigates a distributed model of global evolution of humanity. The model utilizes a onedimensional quasilinear heat equation with a volume source and a nonlinear thermal conductivity coefficient. The model is applied to describe cyclic processes that unfolded over the entire span of human evolution against the backdrop of general population growth with blowup. The model parameters are chosen so that they satisfy the following requirements: the space integral (total population size) increases hyperbolically; the evolution of the system goes through 11 stages corresponding to the main historical epochs in accepted classification and matches actual quantitative indicators.

The asymptotic self-similar behavior for the quasilinear heat equation with nonlinear boundary condition

In this paper the quasilinear heat equation with the nonlinear boundary condition is studied. The blow-up rate and existence of a self-similar solution are obtained. It is proved that the rescaled function v ( y , t ) = ( T − t ) 1 / ( 2 p + α − 2 ) u ( ( T − t ) ( p − 1 ) / ( 2 p + α − 2 ) y , t ) , behaves as t → T like a nontrivial self-similar profile.

Blowup solutions in a problem for the nonlinear heat equation on a small interval

The article considers a one-dimensional quasi-linear heat equation with a volume heat source and a nonlinear thermal conductivity. The analysis is conducted for parameter values where selfsimilar solutions of the equations evolve in an LS-regime with blowup. Heat localization is observed in this case, and the combustion process in the developed stage is in the form of simple or complex structures of contracting half-width. We study the evolution dynamics of various initial distributions and their achievement of the self-similar regime, and also the dependence of the size of the localization region on the shape of the initial compactly supported distribution. The possibility of cyclic evolution of solutions against the background of overall growth with blowup is demonstrated. We particularly focus on the case when the size of the spatial region is much less than the characteristic size of the localization region, and heat flow is obstructed by the physical boundaries. In this case all initial perturbations achieve the self-similar regime, but the corresponding scenario has certain specific features. We present an example of formation of a complex spatial structure that evolves with blowup on a small interval.

Controllability for a parabolic equation with a nonlinear term involving the state and the gradient

In this paper, we prove the controllability of a quasi-linear heat equation involving gradient terms with Fourier boundary conditions in a bounded domain of ℝ N . The proofs of the main results in this paper involve such inequalities and rely on the study of these linear problems and appropriate fixed point arguments.

Continuous dependence results for a class of nonlinear heat equations in a cylindrical region

AbstractIn this paper, we derive bounds for the solutions of a quasilinear heat equation in a finite cylindrical region if the far end and the lateral surface are held at zero temperature, and a nonzero temperature is applied at the near end. Some continuous dependence inequalities are also obtained. We also investigate the case in which a given heat flux is prescribed at the near end, instead of a given temperature. Copyright © 2008 John Wiley & Sons, Ltd.

Optimal Control of Stochastic Partial Differential Equations

We prove a sufficient maximum principle for the optimal control of systems described by a quasilinear stochastic heat equation. The result is applied to solve a problem of optimal harvesting from a system described by a stochastic reaction-diffusion equation.

Identification of a Temperature Dependent Heat Conductivity from Single Boundary Measurements

Considering the identification of a temperature dependent conductivity in a quasilinear elliptic heat equation from single boundary measurements, we proof uniqueness in dimensions $n \ge 2$. Taking noisy data into account, we apply Tikhonov regularization in order to overcome the instabilities. By using a problem-adapted adjoint, we give convergence rates under substantially weaker and more realistic conditions than required by the general theory. Our theory is supported by numerical tests.

A note on blow up of solutions of a quasilinear heat equation with vanishing initial energy

In this work we consider an initial boundary value problem related to the equation u t− div |∇u| m−2∇u =f(u) and prove, under suitable conditions on f, a blow up result for solutions with vanishing or negative initial energy.

Minimal blow-up asymptotics of quasilinear heat equations

We study the asymptotic properties of blow-up solutions u = u(x, t) ≥ 0 of the quasilinear heat equation , where k(u) is a smooth non-negative function, with a given blowing up regime on the boundary u(0, t) = ψ(t) > 0 for t ∈ (0, 1), where ψ(t) → ∞ as t → 1−, and bounded initial data u(x, 0) ≥ 0. We classify the asymptotic properties of the solutions near the blow-up time, t → 1−, in terms of the heat conductivity coefficient k(u) and of boundary data ψ(t); both are assumed to be monotone. We describe a domain, denoted by , of minimal asymptotics corresponding to the data ψ(t) with a slow growth as t → 1− and a class of nonlinear coefficients k(u).We prove that for any problem in S11−, such a blow-up singularity is asymptotically structurally equivalent to a singularity of the heat equation ut = uxx described by its self-similar solution of the form u*(x, t) = −ln(1 − t) + g(ξ), ξ = x/(1 − t)1/2, where g solves a linear ordinary differential equation. This particular self-similar solution is structurally stable upon perturbations of the boundary function and also upon nonlinear perturbations of the heat equation with the basin of attraction .

Multiparameter Fractional Brownian Motion And Quasi-Linear Stochastic Partial Differential Equations

We develop a multiparameter white noise theory for fractional Brownian motion with Hurst multiparameter The theory is used to solve the linear and a quasi-linear heat equation driven by multiparameter fractional white noise. It is proved that for some values of H(depending on the dimension) the solution has a jointly continuous version in t and x

Feedback stabilization of semilinear heat equations

This paper is concerned with the internal and boundary stabilization of the steady‐state solutions to quasilinear heat equations via internal linear feedback controllers provided by an LQ control problem associated with the linearized equation.

On the Spatial Decay of Solutions to a Quasi-Linear Parabolic Initial-Boundary Value Problem and Their Derivatives

In this paper, we study the spatial decay of the solution of a quasi-linear heat equation in a long cylindrical region if the far end and the lateral surface are held at a zero temperature and a nonzero temperature is applied at the near end. Our result follows from the maximum principle applied to an auxiliary function $\Phi$ defined on the solution u and its derivatives.

Incomplete blow-up and singular interfaces for quasilinear heat equations

We study local and asymptotic properties of singular interfaces of the {u=∞} level propagation of the blow-up solutions to the quasilinear heat equation The exponents are chosen in such a particular way because it represents a limit case of the so-called incomplete blow-up where solutions can be continued after blow-up. We consider the proper solutions of the Cauchy problem. We drive the dynamical interface equation and prove that in general singular interfaces are not analytic in time; they are C1-1 and not C2 -functions. The methods apply to other quasilinear equations, for instance to the equation with the $i;p$ei;-Laplacian operator which in the critical incomplete blow-up case has the form We discuss some properties of the blow-up interfaces for equations with general nonlinearities and of the corresponding N-dimensional equations.

Asymptotic self-similar global blow-up for a quasilinear heat equation

We study the asymptotic behavior near finite blow-up time $t = T$ of the solutions to the one-dimensional degenerate quasilinear parabolic equation $$ u_t = (u^\sigma u_x)_x + u^\beta \quad \text{in} \quad \mathbb{R} \times (0,T) ; \quad \sigma>0, \,\, 1<\beta<\sigma+1, $$ with bounded, nonnegative, compactly supported initial data. This parameter range corresponds to global blow-up where $ u(x,t) \to \infty$ as $t \to T^-$ for any $x \in \mathbb{R}$. We prove that the rescaled function $$ f(\xi,t) = (T-t)^{1/(\beta-1)} u(\xi(T-t)^m,t), \quad m = \frac {\beta-(\sigma+1)}{2(\beta-1)} < 0, $$ converges uniformly as $t \to T$ to a unique compactly supported, symmetric self-similar profile $\theta(\xi) \ge 0$ satisfying a nonlinear ordinary differential equation. The proof is based on the intersections comparison (the Sturmian argument) with a two-parametric family of self-similar solutions.

New explicit solutions of quasilinear heat equations with general first-order sign-invariants

For the one-dimensional quasilinear heat equations u t = ψ( u) u xx + f( u), with smooth functions ψ( u) ≥ 0 and f( u), we derive conditions under which a general quadratic Hamilton-Jacobi operator H( u) ≡ u t − [ h 2( u)( u x ) 2 + h 1( u) u x + h 0( u)] (with the functions h 2, h 1, h 0 to be determined) preserves both the signs, ≥ 0 and ≤ 0, on the solutions u( x, t). H( u) is then called a sign-invariant of the given parabolic equation. Several types of different sign-invariants are established which are governed by a linear second-order ordinary differential equations. New explicit solutions via finite-dimensional dynamical systems corresponding to zero-preserving of the first-order operator are constructed.

On symmetric and nonsymmetric blowup for a weakly quasilinear heat equation

We construct blow-up patterns for the quasilinear heat equation $$u_t = \nabla \cdot (k(u)\nabla u) + Q(u)$$ (QHE) in Ω×(0,T), Ω being a bounded open convex set in ℝ N with smooth boundary, with zero Dirichet boundary condition and nonnegative initial data. The nonlinear coefficients of the equation are assumed to be smooth and positive functions and moreoverk(u) andQ(u)/u p with a fixedp>1 are of slow variation asu→∞, so that (QHE) can be treated as a quasilinear perturbation of the well-known semilinear heat equation $$u_t = \nabla u) + u^p .$$ (SHE) We prove that the blow-up patterns for the (QHE) and the (SHE) coincide in a structural sense under the extra assumption $$\smallint ^\infty k(f(e^s ))ds = \infty ,$$ wheref(v) is a monotone solution of the ODEf′(v)=Q(f(v))/v p defined for allv≫1. If the integral is finite then the (QHE) is shown to admit an infinite number of different blow-up patterns.

Blow-Up for Quasilinear Heat Equations Described by Means of Nonlinear Hamilton–Jacobi Equations

It is well-known that the nonnegative solutions of the semilinear heat equation[formula]blow up in a finite timeT(depending on the initial data, assumed to be large enough). This equation is interesting because it exhibits in differentβ-ranges the three most typical blow-up behaviours for solutions of nonlinear parabolic equations. Indeed, we consider radialy symmetric solutions and show that forβ>2 single-point blow-up occurs, forβ<2 blow-up is global, and forβ=2 we have regional blow-up. Moreover, the analysis shows that the precise asymptotic behaviour is described by a nonconstant self-similar blow-up solution of the first-order Hamilton–Jacobi equation[formula]This means that both equations are asymptotically equivalent near blow-up. This type of asymptotic “degeneracy” of a parabolic equation into a first-order equation is actually proved for a more general class of quasilinear heat equations.

On classification of blow-up patterns for a quasilinear heat equation

We study the asymptotic blow-up behavior of nonnegative solutions to the quasilinear heat equation $$ u_t = (u^2)_{xx} + u^2 \quad \text{for $x \in \mathbf{R}, \,\, t > 0$}, $$ with nonnegative, bounded, continuous initial data. We give a complete classification of all possible types of blow-up behavior for compactly supported initial data. For data which look like a step function we construct self-similar blow-up patterns (logarithmic traveling wave solutions) for which the corresponding blow-up sets are empty.