Quasilinear Heat Equation Research Articles

An optimal control problem for single-spot pulsed laser welding

We consider an optimal control problem for a single-spot pulsed laser welding problem. The distribution of thermal energy is described by a quasilinear heat equation. Our emphasis is on materials which tend to suffer from hot cracking when welded, such as aluminum alloys. A simple precursor for the occurrence of hot cracks is the velocity of the solidification front. We therefore formulate an optimal control problem whose objective contains a term which penalizes excessive solidification velocities. The control function to be optimized is the laser power over time, subject to pointwise lower and upper bounds. We describe the finite element discretization of the problem and a projected gradient scheme for its solution. Numerical experiments for material data representing the EN AW 6082-T6 aluminum alloy exhibit interesting laser pulse patterns which perform significantly better than standard ramp-down patterns.

Existence of solution for a class of heat equation with double criticality

In this paper, we study the following class of quasilinear heat equations{ut−ΔΦu=f(x,u)inΩ,t>0,u=0on∂Ω,t>0,u(x,0)=u0(x)inΩ, where ΔΦu=div(φ(x,|∇φ|)∇φ) and Φ(x,s)=∫0|s|φ(x,σ)σdσ is a generalized N-function. We suppose that Ω⊂RN(N≥2) is a smooth bounded domain that contains two open regions ΩN and Ωp with Ω‾N∩Ω‾p=∅. Under some appropriate conditions, the global existence will be done by combining the Galerkin approximations with the potential well theory. Moreover, the large-time behavior of the global weak solution is analyzed. The main feature of this paper consists that −ΔΦu behaves like −ΔNu on ΩN and −Δpu on Ωp, while the continuous function f:Ω×R→R behaves like eα|s|NN−1 on ΩN and |s|p⁎−2s on Ωp as |s|→∞.

A non-conservation stochastic partial differential equation driven by anisotropic fractional Lévy random field

We study a non-conservation second-order stochastic partial differential equation (SPDE) driven by multi-parameter anisotropic fractional Lévy noise (AFLN) and under different initial and/or boundary conditions. It includes the time-dependent linear heat equation and quasi-linear heat equation under the fractional noise as special cases. Unique existence and expressions of solution to the equation are proved and constructed. An AFLN is defined as the derivative of an anisotropic fractional Lévy random field (AFLRF) in certain sense. Comparing with existing noise systems, our non-Gaussian fractional noises are essentially observed from random disturbances on system accelerations rather than from those on system moving velocities. In the process of proving our claims, there are three folds. First, we consider the AFLRF as the generalized functional of sample paths of a pure jump Lévy process. Second, we build Skorohod integration with respect to the AFLN by white noise approach. Third, by combining this noise analysis method with the conventional PDE solution techniques, we provide solid proofs for our claims.

SOLVABILITY OF THE INITIAL-BOUNDARY VALUE PROBLEM FOR THE QUASILINEAR EQUATION OF HEAT CONDUCTIVITY IN DOMAINS THAT CAN BE TRANSFORMED INTO RECTANGLES

In this paper, we study the initial-boundary-value problem for the quasilinear heat equation in regions that are reduced to rectangular. Mathematical modeling of many processes taking place in the real world leads to the study of the problems of equations of mathematical physics, when the areas are not rectangular. The theory of nonlinear problems is an actively developing section of the theory of modern differential equations. In the theory of nonlinear equations, a special place is occupied by the study of unbounded solutions or, in other words, modes with exacerbation. Nonlinear evolutionary problems that allow unlimited solutions are globally unsolvable: solutions grow unlimitedly over a finite period of time. In this paper, the initial-boundary-value problem for the quasilinear heat equation in regions that can be reduced to rectangular ones, the existence of a solution is proved by the Galerkin method. The uniqueness of the solution was proved by the obtained a priori estimates. Sufficient conditions for the destruction of the solution in a finite time in a bounded domain are obtained. The exponential decay of the solution with an infinite increase in time is proved. In the final time, it was proved that the solution is localized, i.e. disappears (nullifies).

Nondivergence form quasilinear heat equations driven by space-time white noise

We give a Wong-Zakai type characterisation of the solutions of quasilinear heat equations driven by space-time white noise in 1+1 dimensions. In order to show that the renormalisation counterterms are local in the solution, a careful arrangement of a few hundred terms is required. The main tool in this computation is a general ‘integration by parts’ formula that provides a number of linear identities for the renormalisation constants.

Parallel algorithm for numerical solution of heat equation in complex cylindrical domain

In this article we present a parallel algorithm for simulation of the heat conduction process inside the so-called pulse cryogenic cell. This simulation is important for designing the device for portion injection of working gases into ionization chamber of ion source. The simulation is based on the numerical solving of the quasilinear heat equation with periodic source in a multilayered cylindrical domain. For numerical solution the Alternating Direction Implicit (ADI) method is used. Due to the non-linearity of the heat equation the simple-iteration method has been applied. In order to ensure convergence of the iteration process, the adaptive time-step has been implemented. The parallelization of the calculation has been realized with shared memory application programming interface OpenMP and the performance of the parallel algorithm is in agreement with the case studies in literature.

SPDEs with fractional noise in space: Continuity in law with respect to the Hurst index

In this article, we consider the quasi-linear stochastic wave and heat equations on the real line and with an additive Gaussian noise which is white in time and behaves in space like a fractional Brownian motion with Hurst index $H\in (0,1)$. The drift term is assumed to be globally Lipschitz. We prove that the solution of each of the above equations is continuous in terms of the index $H$, with respect to the convergence in law in the space of continuous functions.

Grow-up for a quasilinear heat equation with a localized reaction

We study the behaviour of global solutions to the quasilinear heat equation with a reaction localizedut=(um)xx+a(x)up,m,p>0 and a(x) being the characteristic function of an interval. We prove that there exists an exponent p0=max⁡{1,m+12} such that all global solutions are bounded if p>p0, while for p≤p0 all the solutions are global and unbounded. In the last case, we prove that if p<m the grow-up rate is different to the one obtained when a(x)≡1, while if p>m the grow-up rate coincides with that rate, but only inside the support of a; outside the interval the rate is smaller.

A Distributed-Parameter Approach for the Surface Temperature Estimation of an LED Heated Silicon Wafer

A variety of processes in the semiconductor industry require heating of the silicon wafer to the desired temperature. This process is widely referred to as rapid thermal processing. However, the contactless measurement of the surface temperature of the wafer is still a major challenge, especially in the case of rotating wafers. Measurements using infrared cameras are not suitable due to the fact that silicon is transparent in this wavelength range. Special sensors based on the principle of pyrometry are available, but such sensors can only measure the surface temperature at one single point. This paper presents an observer approach that estimates the wafer’s surface temperature by using the temperature measurement of only one pyrometer. The approach is based on a mathematical model capturing the dynamical behavior of the wafer’s temperature. It relies mainly on the quasi-linear heat equation. Real world experiments demonstrate the achieved accuracy of the proposed approach.

English

english

Grow-up for a quasilinear heat equation with a localized reaction in higher dimensions

We study the behaviour of nonnegative global solutions to the quasilinear heat equation with a reaction localized in a ball $$\begin{aligned} u_t={\varDelta } u^m+a(x)u^p, \end{aligned}$$ for $$m>0$$ , $$0<p\le \max \{1,m\}$$ , $$a(x)=\mathbb {1}_{B_L}(x)$$ , $$0<L<\infty $$ and $$N\ge 2$$ . We study when the solutions are bounded or unbounded. In particular we show that the precise value of the length L plays a crucial role in the critical case $$p=m$$ for $$N\ge 3$$ . We also obtain the asymptotic behaviour of unbounded solutions and prove that the grow-up rate is different in most of the cases to the one obtained when $$L=\infty $$ .

A discrete optimality system for an optimal harvesting problem

In this paper, we obtain the discrete optimality system of an optimal harvesting problem. While maximizing a combination of the total expected utility of the consumption and of the terminal size of a population, as a dynamic constraint, we assume that the density of the population is modeled by a stochastic quasi-linear heat equation. Finite-difference and symplectic partitioned Runge–Kutta (SPRK) schemes are used for space and time discretizations, respectively. It is the first time that a SPRK scheme is employed for the optimal control of stochastic partial differential equations. Monte-Carlo simulation is applied to handle expectation appearing in the cost functional. We present our results together with a numerical example. The paper ends with a conclusion and an outlook to future studies, on further research questions and applications.

The dual approach to stationary and evolution quasilinear PDEs

In this paper, we study several quasilinear PDEs with a particular algebraic structure. In the case of stationary solutions for a quasilinear Schrödinger equation, Colin and Jeanjean (Nonlinear Anal 56:213–226, 2004) implemented a dual approach to prove existence and qualitative properties of the solutions. The method takes advantage of the particular underlying structure of that quasilinear PDE, which is essentially semilinear. The main goal of this manuscript is to show that the dual approach is successful in a broader set of problems, especially in the stationary cases involving more general quasilinear terms and the p-Laplacian. We prove also that this approach works for some quasilinear heat equations, but fails for the complete evolution of quasilinear Schrödinger equations. The reason of the failure seems related to the extra structure of the complex plane.

Determination of a coefficient in a quasilinear parabolic equation with periodic boundary condition

In this paper, we consider the problem of determining the time-dependent thermal diffusivity and the temperature distribution in one-dimensional quasilinear heat equation with periodic boundary and integral overdetermination conditions. We establish conditions for the existence, uniqueness and continuous dependent of a classical solution of the problem under considerations. We present some results on the numerical solution with two examples.

Invariance of convex sets for non-autonomous evolution equations governed by forms

We consider a non-autonomous form a : [ 0 , T ] × V × V → C where V is a Hilbert space which is densely and continuously embedded in another Hilbert space H. Denote by A ( t ) ∈ L ( V , V ' ) the operator associated with a ( t , · , · ) . Given f ∈ L 2 ( 0 , T , V ' ) , one knows that for each u 0 ∈ H there is a unique solution u ∈ H 1 ( 0 , T ; V ' ) ∩ L 2 ( 0 , T ; V ) of u ˙ ( t ) + A ( t ) u ( t ) = f ( t ) , u ( 0 ) = u 0 . This result by J. L. Lions is well known. The aim of this article is to find a criterion for the invariance of a closed convex subset C of H; that is, we give a criterion on the form which implies that u ( t ) ∈ C for all t ∈ [ 0 , T ] whenever u 0 ∈ C . In the autonomous case for f = 0 , the criterion is known and even equivalent to invariance by a result proved by Ouhabaz ‘Invariance of closed convex sets and domination criteria for semigroups’, Potential Anal. 5 (1996) 611–625. See also Ouhabaz ‘Analysis of heat equations on domains’, London Mathematical Society Monographs. Princeton University Press, Princeton, NJ, 2005. We give applications to positivity and comparison of solutions to heat equations with non-autonomous Robin boundary conditions. We also prove positivity of the solution to a quasi-linear heat equation.

WEAK CONVERGENCE FOR QUASILINEAR STOCHASTIC HEAT EQUATION DRIVEN BY A FRACTIONAL NOISE WITH HURST PARAMETER H ∈ (½, 1)

In this paper, we consider a quasi-linear stochastic heat equation in one dimension on [0, 1], with Dirichlet boundary conditions driven by an additive fractional white noise. We formally replace the random perturbation by a family of noisy inputs depending on a parameter n ∈ ℕ which can approximate the fractional 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 [Formula: see text] of continuous functions, to the solution of the fractional noise driven SPDE.

Weak approximation of the fractional Brownian sheet from random walks

In this paper, we show an approximation in law of the fractional Brownian sheet by random walks. As an application, we consider a quasilinear stochastic heat equation with Dirichlet boundary conditions driven by an additive fractional noise.

Sign-changing solutions of a quasilinear heat equation with a source term

The Cauchy problem of a heat equation with a source term $$ \psi_t=\Delta \left(|\psi|^{m-1}\psi\right)+|\psi|^{\gamma-1}\psi\ \ \mbox{in}\ \ (0, \infty)\times R^n $$ is considered, where $\gamma>m>1$. We are interested in global solutions with H$\ddot{o}$lder continuity which satisfy the equation in the distribution sense, and with a fixed number of sign changes at any given time $ t > 0$. Through detailed analysis of the self-similarity problem, we prove the existence of two type of such solutions, one with compact support and the other decays to zero as $ | x| \rightarrow \infty$ with an algebraic rate determined uniquely by $ n, m$ and $\gamma$. Our results extend previous study on positive self-similar solutions. Moreover, they demonstrate vital difference from the well-studied semi-linear case of $m = 1$.

Damped-Driven KdV and Effective Equations for Long-Time Behaviour of its Solutions

For the damped-driven KdV equation $$ \dot{u}-{\nu}u_{xx} + u_{xxx} - 6uu_{x} = \sqrt{\nu}\,\eta(t, x), x \in S^1, \int udx \equiv \int \eta dx \equiv 0, $$ with 0 < ν ≤ 1 and smooth in x white in t random force η, we study the limiting long-time behaviour of the KdV integrals of motions (I1, I2, . . . ), evaluated along a solution uν(t, x), as ν → 0. We prove that for \({0 \leq \tau := {\nu}t \lesssim 1}\) the vector Iν(τ) = (I1(uν(τ, ·)), I2(uν(τ, ·)), . . . ), converges in distribution to a limiting process \({I^{0}(\tau) = (I_{1}^{0}, I_{2}^{0}, \ldots )}\). The j-th component \({I_{j}^{0}}\) equals \({\frac{1}{2}(v_{j}(\tau)^2 + v_{-j}(\tau)^{2})}\), where the vector v(τ) = (v1(τ), v−1(τ), v2(τ), . . . ) is a solution of a system of effective equations for the damped-driven KdV. These new equations are a quasilinear stochastic heat equation with a non-local nonlinearity, written in the Fourier coefficients. They are well posed.

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

In this article, we consider the controllability of a quasi-linear heat equation involving gradient terms with Dirichlet boundary conditions in a bounded domain of R N . The results are established by using the variational methods, the related duality theory and Kakutani Fixed-point Theorem.