Global dynamics for the energy-critical nonlinear heat equation

A nonlinear forward-backward heat equation with a regularization term was proposed by Barenblatt et al. [1, 2] to model the heat and mass exchange in stably stratified turbulent shear flow. It was proven to be well-posed in the case of given initial and Neumann boundary conditions. However, the solution was found to have an unphysical discontinuity with certain smooth initial functions. In this paper, a nonlinear heat equation with a time delay originally used by Barenblatt et al. [1, 2] to derive their model is investigated. The same type of initial-boundary value problem is shown to have a unique smooth global solution when the initial function is reasonably smooth. Numerical examples are used to demonstrate that its solution forms step-like profiles in finite times. A semi-discretization of the initial-boundary value problem is proved to have a unique asymptotically and globally stable equilibrium.

We consider the nonlinear half-Laplacian heat equation $$\begin{aligned} u_t+(-\Delta )^{\frac{1}{2}} u-|u|^{p-1}u=0,\quad {\mathbb {R}}^n\times (0,T). \end{aligned}$$We prove that all blows-up are type I, provided that \(n \le 4\) and \( 1<p<p_{*} (n)\) where \( p_{*} (n)\) is an explicit exponent which is below \(\frac{n+1}{n-1}\), the critical Sobolev exponent. Central to our proof is a Giga-Kohn type monotonicity formula for half-Laplacian and a Liouville type theorem for self-similar nonlinear heat equation. This is the first instance of a monotonicity formula at the level of the nonlocal equation, without invoking the extension to the half-space.

A simple algorithm for solving Cauchy problem of nonlinear heat equation without initial value

A White Noise Approach to a Class of Non-linear Stochastic Heat Equations

Exact solutions of linear and non-linear differential-difference heat and diffusion equations with finite relaxation time

We study nonlinear wave and heat equations on ℝ d driven by a spatially homogeneous Wiener process. For the wave equation we consider the cases of d = 1, 2, 3. The heat equation is considered on an arbitrary ℝ d -space. We give necessary and sufficient conditions for the existence of a function-valued solution in terms of the covariance kernel of the noise.

We formulate and study a one–dimensional single–species diffusive–delay population model. The time delay is the time taken from birth to maturity. Without diffusion, the delay differential model extends the well–known logistic differential equation by allowing delayed constant birth processes and instantaneous quadratically regulated death processes. This delayed model is known to have simple global dynamics similar to that of the logistic equation. Through the use of a sub/supersolution pair method, we show that the diffusive delay model continues to generate simple global dynamics. This has the important biological implication that quadratically regulated death processes dramatically simplify the growth dynamics. We also consider the possibility of travelling wavefront solutions of the scalar equation for the mature population, connecting the zero solution of that equation with the positive steady state. Our main finding here is that our fronts appear to be all monotone, regardless of the size of the delay. This is in sharp contrast to the frequently reported findings that delay causes a loss of monotonicity, with the front developing a prominent hump in some other delay models.

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.

We prove several differential Harnack inequalities for positive solutions to nonlinear backward heat equations with different potentials coupled with the Ricci flow. We also derive an interpolated Harnack inequality for the nonlinear heat equation under the $\varepsilon$-Ricci flow on a closed surface. These new Harnack inequalities extend the previous differential Harnack inequalities for linear heat equations with potentials under the Ricci flow.

A parameter‐dependent perturbation of the spectrum of the scalar Laplacian is studied for a class of nonlocal and non‐self‐adjoint rank one perturbations. A detailed description of the perturbed spectrum is obtained both for Dirichlet boundary conditions on a bounded interval as well as for the problem on the full real line. The perturbation results are applied to the study of a related parameter‐dependent nonlinear and nonlocal parabolic equation. The equation models a feedback system that admits an interpretation as a thermostat device or in the context of an agent‐based price formation model for a market. The existence and the stability of periodic self‐oscillations of the related nonlinear and nonlocal heat equation that arise from a Hopf bifurcation are proved. The bifurcation and stability results are obtained both for the nonlinear parabolic equation with Dirichlet boundary conditions and for a related problem with nonlinear Neumann boundary conditions that model feedback boundary control. They follow from a Popov criterion for integral equations after reducing the stability analysis for the nonlinear parabolic equation to the study of a related nonlinear Volterra integral equation. While the problem is studied in the scalar case only, it can be extended naturally to arbitrary Euclidean dimension and to manifolds.

Application of homotopy-perturbation and variational iteration methods to nonlinear heat transfer and porous media equations

This paper considers the inverse determination of the positive unknown thermal properties K(T), C(T) and the unknown temperature T(x,t) in the nonlinear transient heat conduction equation. In addition to prescribed initial and/or boundary values, specified continuously differentiable temperature data T(x0, t) with non-zero derivative at a single sensor location x = x0 is given. When K(T) and C(T) obey a certain relationship which enables one to linearise exactly the nonlinear heat equation then their dependence upon T is obtained explicitly, whilst the unknown temperature T(x,t) is obtained implicitly and is then calculated numerically. Results are presented and discussed for infinite, semi-infinite and finite slabs.

On symmetry reduction and invariant solutions to some nonlinear multidimensional heat equations

In this paper, we consider the (2+1) nonlinear fractional heat equation with non-local integral terms and investigate two different cases of such non-local integral terms. The first has to do with the time-dependent non-local integral term and the second is the space-dependent non-local integral term. Apart from the nonlinear nature of these formulations, the complexity due to the presence of the non-local integral terms impelled us to use a relatively new analytical technique called q-homotopy analysis method to obtain analytical solutions to both cases in the form of convergent series with easily computable components. Our numerical analysis enables us to show the effects of non-local terms and the fractional-order derivative on the solutions obtained by this method.

In 2016 the authors performed space experiments to obtain the tropospheric NO2 field with the horizontal spatial resolution for the first time at the world level reaching 2.4 km. The NO2 fields were restored based on spectral measurements of GSA instrument installed on board the Russian satellites of the Resurs-P series. For the first time, the high spatial resolution of the new method makes it possible to identify local sources of NO2 pollution and their plumes. Good agreement with OMI NO2 observations with resolution 13 km x 24 km confirmed the reliability of the obtained Resurs-P NO2 fields in general. For the validation of high-detailed structures detected in the NO2 fields of GSA/Resurs-P, we are developing methods based on comparisons with chemical transport models. The comparison is performed for Hebei province, the North China Plain, which is the most NO2 polluted area in the world, using Resurs-P data obtained on September 29, 2016. The paper presents preliminary comparison of the Resurs-P tropospheric NO2 field with simulation based on HYSPLIT transport model. For the solution of the problem a high-detailed chemical transport model based on a solution of the nonlinear heat and mass transfer equation is under development. A theoretical background of the methods of asymptotic analysis of multidimensional singularly perturbed problems for the nonlinear heat and mass transfer equation is proposed.