Nonlinear Heat Equation Research Articles

On type I blowup of some nonlinear heat equations with a potential

In this paper, we are concerned with the following initial-boundary value problem{ut=Δu+Q(|x|)|u|p−1u,x∈BR,t>0u(x,t)=0,x∈∂BR,t>0u(x,0)=u0(x),x∈BR, where p≥ps:=N+2N−2, u0∈L∞(BR), and Q(r)∈C1([0,R]), 0<C_≤Q(r)≤C‾<∞,Q′(r)≤0. We extend the asymptotic behavior results, which is well-known when Q is constant according to Matano-Merle (cf. [25]), for the blow-up solutions. More precisely, we show that when ps≤p<p⁎, the blowup of radial solution to this problem is always of Type I. This result partially generalizes the conclusions in [25] for Q≡1. This extension is nontrivial due to the appearance of Q. The quasi-monotonicity formula established by the third author and Cheng in [8] allows us to use an energy method to get a priori estimates on the rescaled solutions. The contraction mapping principle shows the existence of singular stationary solutions to an associated elliptic equation with a potential. In the end, the properties of zero number for solutions lead to the nonexistence of type II singularity for the problem.

A remark on randomization of a general function of negative regularity

In the study of partial differential equations (PDEs) with random initial data and singular stochastic PDEs with random forcing, we typically decompose a classically ill-defined solution map into two steps, where, in the first step, we use stochastic analysis to construct various stochastic objects. The simplest kind of such stochastic objects is the Wick powers of a basic stochastic term (namely a random linear solution, a stochastic convolution, or their sum). In the case of randomized initial data of a general function of negative regularity for studying nonlinear wave equations (NLW), we show necessity of imposing additional Fourier–Lebesgue regularity for constructing Wick powers by exhibiting examples of functions slightly outside L 2 ( T d ) L^2(\mathbb {T}^d) such that the associated Wick powers do not exist. This shows that probabilistic well-posedness theory for NLW with general randomized initial data fails in negative Sobolev spaces (even with renormalization). Similar examples also apply to stochastic NLW and stochastic nonlinear heat equations with general white-in-time stochastic forcing, showing necessity of appropriate Fourier–Lebesgue γ \gamma -radonifying regularity in the construction of the Wick powers of the associated stochastic convolution.

Towards a theory of steady-state solidification process with a quasi-equilibrium two-phase region

AbstractThe process of directional crystallization in the presence of a quasi-equilibrium two-phase region located between the solid material and the liquid phase is studied theoretically. The mathematical model of the process is based on heat and mass transfer equations in the solid, liquid and two-phase regions, as well as boundary conditions at the phase interfaces “solid phase” – “two-phase region” and “two-phase region” – “liquid phase”, which are moving with a constant velocity. The process of directional crystallization is given by fixed temperature gradients in the solid and liquid phases, which determine a constant velocity of melt solidification. An exact analytical solution of the nonlinear problem with two moving boundaries of phase transformation is obtained, which is based on the transition to a new independent variable, the solid phase fraction, when integrating the nonlinear heat and mass transfer equations in the two-phase region. As a result of solving the problem, the distributions of temperature and concentration of dissolved impurity, the solid phase fraction in a two-phase region, the laws and velocities of motion of its interphase boundaries are determined. It is analytically shown that the impurity concentration and temperature in the two-phase region are only the functions of solid phase fraction, which, in turn, depends on the spatial coordinate. Analysis of the obtained solutions shows that the solid phase fraction in a two-phase region can be both a decreasing and increasing function of the spatial coordinate, which is directed from the solid material to the melt. This determines the internal structure of two-phase region, its permeability, average interdendritic spacing, distribution of dissolved impurity, crystallization velocity and laws of two-phase region boundaries.

Multiscale mushy layer model for Arctic marginal ice zone dynamics

Perhaps the most dynamic component of the Arctic sea ice cover is the marginal ice zone (MIZ), the transitional region between dense pack ice to the north and open ocean to the south. It widens by a factor of four while seasonally migrating more than 1600 km poleward in the Bering–Chukchi Sea sector, impacting climate dynamics, ecological processes, and human accessibility to the Arctic. Here we showcase a transformative mathematical modeling approach to understanding changes in MIZ location and width, focusing on their seasonal cycles as observed by satellites. We view the MIZ as a liquid-solid phase transition region, or mushy zone, on the scale of the Arctic Ocean. Invoking the physics of phase changes, the MIZ is modeled as a dynamic, multiscale composite material layer; this model captures 96% of the annual cycle of MIZ location and 78% of the annual cycle of MIZ width. Temperature in the upper ocean is described by a nonlinear heat equation with effective parameters obtained using homogenization theory for a random medium of ice floes in a sea water host. Observations and simulations together indicate that MIZ location closely tracks the below-ice 273 K isotherm while the width of the MIZ follows vertical heat flux convergence, but with a three-week lag.

Complex-plane singularity dynamics for blow up in a nonlinear heat equation: analysis and computation

Abstract Blow-up solutions to a heat equation with spatial periodicity and a quadratic nonlinearity are studied through asymptotic analyses and a variety of numerical methods. The focus is on tracking the dynamics of the singularities in the complexified space domain all the way from the initial time until the blow-up time, which occurs when the singularities reach the real axis. This widely applicable approach gives forewarning of the possibility of blow up and an understanding of the influence of singularities on the solution behaviour on the real axis, aiding the (perhaps surprisingly involved) asymptotic analysis of the real-line behaviour. The analysis provides a distinction between small and large nonlinear effects, as well as insight into the various time scales over which blow up is approached. The solution to the nonlinear heat equation in the complex spatial plane is shown to be related asymptotically to a nonlinear ordinary differential equation. This latter equation is studied in detail, including its computation on multiple Riemann sheets, providing further insight into the singularities of blow-up solutions of the nonlinear heat equation when viewed as multivalued functions in the complex space domain and illustrating the potential intricacy of singularity dynamics in such (non-integrable) nonlinear contexts.

Exact Solutions with Generalized Separation of Variables of the Nonlinear Heat Equation with a Source

Exact Solutions with Generalized Separation of Variables of the Nonlinear Heat Equation with a Source

Numerical analysis of the Maxwell-Cattaneo-Vernotte nonlinear model

In the literature, one can find numerous modifications of Fourier’s law, the first of which is the Maxwell–Cattaneo–Vernotte heat equation. Although this model has been known for decades and successfully used to model low-temperature damped heat wave propagation, its nonlinear properties are rarely investigated. This paper aims to present the functional relationship between the transport coefficients and the consequences of their temperature dependence, particularly focusing on thermal conductivity. These are the consequences of the second law of thermodynamics. Furthermore, we introduce a particular implicit numerical scheme in order to solve such nonlinear heat equations reliably, free from artificial numerical errors. We investigate the scheme’s stability, dissipation, and dispersion attributes. We demonstrate the effect of temperature-dependent thermal conductivity on two different initial-boundary value problems, including time-dependent boundaries and heterogeneous initial conditions, for which we discuss the nontrivial constraints following irreversible thermodynamics.

The inverse problem of heat conduction in the case of non-uniqueness: A functional identification approach

Abstract A problem of identification of the set of a thermal-conductivity coefficients for the nonlinear heat equation in the case of non-uniqueness is considered. Classes of inverse heat conduction problems (IHCP) with a non-unique solution are defined. Explicit descriptions of sets of thermal-conductivity coefficients for these classes are obtained. For solving the identification problem the functional identification approach is used. Unlike traditional methods, the proposed algorithm does not utilize approximations of the coefficient with a finite system of basis functions. The results of computational experiments are presented. It is shown that the functional identification approach makes it possible to numerically identify the non-uniqueness of the solution of the inverse problem of heat conduction.

Two-Sided Estimates of Solutions with a Blow-Up Mode for a Nonlinear Heat Equation with a Quadratic Source

Two-Sided Estimates of Solutions with a Blow-Up Mode for a Nonlinear Heat Equation with a Quadratic Source

Transient Heat Transfer Analysis in Metal Plates with Variable Thickness

Nonlinear transient heat transfer via conduction–radiation is a dynamic topic of long-standing interest with applications ranging from aeronautical and mechanical engineering to industrial and civil security. To gain a better understanding of the performances of materials having thermal proprieties that change during nonlinear heat transfer, several studies using the finite element method (FEM) have been conducted. Such studies apply nonlinear thermal material characteristics to describe the complete system under different loading conditions in each region by adjusting the temperature values for the other three edges and the thickness parameter with Dirichlet boundary conditions. As a result, while modeling and simulating temperature distributions for such situations, nonlinearities generated by temperature-dependent thermal conductivity must be considered. In this work, we focus on the analysis of coupled transient heat transfer through two metal plates with temperature-dependent thermal characteristics in which the temperature is fixed along the bottom edge and heat is transferred from both the top and bottom faces of the two plates. FEM is employed to solve the nonlinear heat equation and compute the temperature as a function of time for variable thickness. The study examines the effect of modifying the thickness parameter values on the temperature distribution over time for various edge values over 5,000 s.

Fractional nonlinear heat equations and characterizations of some function spaces in terms of fractional Gauss–Weierstrass semi–groups

AbstractWe present a new proof of the caloric smoothing related to the fractional Gauss–Weierstrass semi–group in Triebel-Lizorkin spaces. This property will be used to prove existence and uniqueness of mild and strong solutions of the Cauchy problem for a fractional nonlinear heat equation.

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We consider a nonlinear parabolic equation with a nonlocal term which preserves the L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document}-norm of the solution. We study the local and global well-posedness on a bounded domain, as well as the whole Euclidean space, in H1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^1$$\\end{document}. Then we study the asymptotic behavior of solutions. In general, we obtain weak convergence in H1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^1$$\\end{document} to a stationary state. For a ball, we prove strong convergence to the ground state when the initial condition is positive.

Positional embeddings for solving PDEs with evolutional deep neural networks

This work extends the paradigm of evolutional deep neural networks (EDNNs) to solving parametric time-dependent partial differential equations (PDEs) on domains with geometric structure. By introducing positional embeddings based on eigenfunctions of the Laplace-Beltrami operator, geometric properties are encoded intrinsically and Dirichlet, Neumann and periodic boundary conditions of the PDE solution are enforced directly through the neural network architecture. The proposed embeddings lead to improved error convergence for static PDEs and extend EDNNs towards computational domains of realistic complexity.Several steps are taken to improve performance of EDNNs: Solving the EDNN update equation with a Krylov solver avoids the explicit assembly of Jacobians and enables scaling to larger neural networks. Computational efficiency is further improved by an ad-hoc active sampling scheme that uses the PDE dynamics to effectively sample collocation points. A modified linearly implicit Rosenbrock method is proposed to alleviate the time step requirements of stiff PDEs. Lastly, a completely training-free approach, which automatically enforces initial conditions and only requires time integration, is compared against EDNNs that are trained on the initial conditions. We report results for the Korteweg-de Vries equation, a nonlinear heat equation and (nonlinear) advection-diffusion problems on domains with and without holes and various boundary conditions, to demonstrate the effectiveness of the method. The numerical results highlight EDNNs as a promising surrogate model for parametrized PDEs with slow decaying Kolmogorov n-width.

Challenges in Numerical Solutions of Higher-Dimensional Differential Equations

Differential equations constitute a fundamental tool in modeling various natural phenomena across scientific disciplines such as physics, engineering, and finance. We provide an overview of fractional differential equations, focusing on the computational requirements associated with their numerical solutions from a computer science perspective. We analyze the computational intricacies concerning First-Order Linear ODE, First-Order Nonlinear ODE, Second-Order Linear ODE, Second-Order Nonlinear ODE, Heat Equation (PDE), and Wave Equation (PDE). This comparative assessment delves into the computational demands of solving these equations using differential equation methodologies. While analytical solutions provide deep insights, obtaining numerical solutions, particularly in higher dimensions, remains a persistent challenge. Finite difference methods commonly employed for numerical solutions, In higher-dimensional problems, traditional numerical methods face challenges stemming from an exponential surge in grid points and the consequent demand for substantially decreased time step sizes. This paper explores the challenges posed by higher-dimensional differential equations in numerical solutions. It highlights the infeasibility of finite difference methods in such scenarios and emphasizes the need for innovative numerical techniques capable of efficiently handling the complexities of higher-dimensional differential equations. Overcoming these challenges is crucial for advancing our understanding and modeling capabilities in complex real-world systems governed by differential equations. Continued research efforts strive to develop novel numerical methodologies capable of addressing these challenges, aiming to broaden the scope of solvable higher-dimensional differential equations and expand their application across diverse scientific domains.

Decay rates for the 4D energy‐critical nonlinear heat equation

AbstractIn this paper, we address the decay of solutions to the four‐dimensional energy‐critical nonlinear heat equation in the critical space . Recently, it was proven that the norm of solutions goes to zero when time goes to infinity, but no decay rates were established. By means of the Fourier Splitting Method and using properties arising from the scale invariance, we obtain an algebraic upper bound for the decay rate of solutions.

Invariant Measures for the Nonlinear Stochastic Heat Equation with No Drift Term

Invariant Measures for the Nonlinear Stochastic Heat Equation with No Drift Term

Global solution of nonlinear Heat equation with solutions in a Hilbert Manifold

The objective of this paper is to deal with the deterministic problem consisting of non-linear heat equation of gradient type. The evolution equation emerges as projecting the Laplace operator with Dirichlet boundary conditions and polynomial nonlinearity of degree 2n−1, onto the tangent space of a sphere M in a Hilbert space ℋ. We are going to deal with questions of the existence and the uniqueness of a global solution, and the invariance of the manifold, M i.e. if the suitable initial data lives on M then all trajectories of solutions also belong to M. Finally, we will also show the global solution generates a gradient flow.

Solution to a Two-Dimensional Nonlinear Parabolic Heat Equation Subject to a Boundary Condition Specified on a Moving Manifold

Solution to a Two-Dimensional Nonlinear Parabolic Heat Equation Subject to a Boundary Condition Specified on a Moving Manifold

Standing-sphere blow-up solutions for the nonlinear heat equation

Standing-sphere blow-up solutions for the nonlinear heat equation

Heat treatment of ground surface using an electric arc plasma generator

The article presents a generalization of the results of experimental and theoretical studies on the intensive thermal effect on the surface of foundations from cohesive soils (as opposed to deep heat treatment) using an electric arc generator of low-temperature plasma (plasmatron). A plasmatron with massive consumable electrodes made of borosilicated graphite and a wide "blurred" plasma torch was used for heat treatment. Physical and mathematical modeling of the thermal effect on the surface of clay soil by a moving high-temperature source has been performed. For the first time, a special installation was manufactured and experimental studies were carried out on the heat treatment of ground surfaces on the site with an electric arc plasma torch. It is established that the effective heat flux density reaches 2.5-.3.5 kW/cm2. The main part of the thermal energy for the used plasmatron enters the material due to radiant heat transfer. An analytical mathematical model of the temperature field of a three-dimensional semi-bounded body in the technological process under consideration is given by an essentially nonlinear boundary value problem for a nonlinear heat equation with nonlinear boundary conditions of the second kind. When solving thermophysical problems, the ground semi-bounded space was considered as a quasi-homogeneous medium having a constant initial temperature and initial thermophysical parameters that change in the process of temperature increase. The effective values of the thermophysical characteristics were determined on the basis of experimental data. Calculations and experiments have shown that the temperature boundary leading to significant changes in the structural properties of soils drops to a depth of 4-6 cm from the surface, despite the boiling of the ground melt with a temperature of 2500 ...2800 K on the ground surface. Therefore, a new technological principle of heat treatment is proposed, which consists in building up the ground melt in layers of 4-5 cm up from the initial surface. The impossibility of obtaining a positive effect even with the use of a powerful high-temperature source of exposure in the case of traditional surface heat treatment technology has been confirmed. The new plasma technology of surface heat treatment of ground surfaces to the stage of silicate melt allows to obtain a common layer of the required thickness. At the same time, the efficiency of surface heat treatment is significantly increased. The new technology of layer-by-layer surface deposition reduces cracking in the layer when the melt cools, but does not eliminate this negative process.