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We prove that generating series for colored modular graphs satisfy some systems of partial differential equations generalizing Burgers or heat equations. The solution is obtained by genus expansion of the generating function. The initial term of this expansion is the corresponding generating function for trees. For this term the system of differential equations is equivalent to the inversion problem for the gradient mapping defined by the initial condition. This enables to state the Jacobian conjecture in the language of generating functions. The use of generating functions provides rather short and natural proofs of resent results of Zhao and of the well-known Bass–Connell–Wright tree inversion formula.

Motivated by the paper Rosenbloom–Widder [A temperature function which vanishes initially, Amer. Math. Mon.65(8) (1958) 607–609], we construct a more general family of nonzero solutions to the $1$-dimensional heat equation $u_t=u_{xx}$ on ${ℝ}^2$ with zero initial data. These solutions are analytically more manageable than the well-known Tychonoff power series solution. Nonzero solutions to the $n$-dimensional heat equation $u_t=△u$ on ${ℝ}^{n+1}$ with zero initial data (i.e. vanishing on the hyperplane $P:\{t=0\}$) can be easily obtained as a consequence of the examples on ${ℝ}^2.$ Finally, given an arbitrary hyperplane $P\subset {ℝ}^{n+1}$, we can construct a family of nonzero solutions which vanish on $P\subset {ℝ}^{n+1}.$

In this paper, we consider a nonlinear control problem for one-dimensional viscous Burgers’ equation associated with a controlled linear heat equation by means of the Hopf–Cole transformation. The control is carried out by the time-dependent intensity of a distributed heat source influencing the heat equation. The set of admissible controls consists of compactly supported $L^{\infty }$ functions. Using the Green’s function approach, we analyze the possibilities of exact and approximate establishment of a given terminal state for the associated nonlinear Burgers’ equation within a desired amount of time. It is shown that the exact controllability of the associated Burgers’ equation and the heat equation are equivalent. Furthermore, sufficient conditions for the approximate controllability are derived. The set of resolving controls is constructed in both cases. The determination of the resolving controls providing exact controllability is reduced to an infinite-dimensional system of linear algebraic equations. By means of the heuristic method of resolving control determination, parametric hierarchies of solutions providing approximate controllability are constructed. The results of a numerical simulation supporting the theoretical derivations are discussed.

In this paper, a solution to heat equation in a spherical model is proposed. The model takes into account conjointly the boundary conditions and the transfer equations in a spray device. The deposited layer is supposed to be Gaussian and not uniform like most recent studies. The solution, using Legendre, Bessel and Boubaker polynomials expansions, presents the advantage of being continuous and indefinitely derivable. These features make the resolution connectable to any heat transfer similar model.

Multi-wave resonant solutions of a class of nonlinear heat-type equations are investigated by their corresponding generalized bilinear equations. We develop a new technique for searching for resonant solutions to those generalized bilinear equations, using the idea of weights of dependent variables. The results show that generalized bilinear operators and generalized bilinear equations are powerful and irreplaceable tools for dealing with nonlinear differential equations.

A thermo-viscoelastic contact problem with Coulomb friction is studied. The model involves a temperature-dependent heat conductivity with moderate growth in order to control the rapidly growing mixed terms as e.g. the heat generated by friction or by viscous deformation. The contact condition is formulated in velocities. The existence of a solution is proved by an approximation of the problem in several steps involving a penalty-approximation and a smoothing of the friction law. The solvability of the approximate problem is proved by the Galerkin method. A priori estimates uniform with respect to all approximation parameters make it possible to pass to the original problem. These estimates are based on a regularity result for the contact problem without heat transfer.

We analyze the linear theory of parabolic equations in uniform spaces. We obtain sharp L^p-L^q-type estimates in uniform spaces for heat and Schrödinger semigroups and analyze the regularizing effect and the exponential type of these semigroups. We also deal with general second-order elliptic operators and study the generation of analytic semigoups in uniform spaces.

Existence of weak solutions is proved for a system of nonlinear parabolic equations/inequalities describing evolution of magnetization, temperature, magnetic field, and electric field in electrically-conductive unsaturated ferromagnets. The system is derived from a recently-proposed thermodynamically-consistent continuum theory for the ferro/paramagnetic transition. Besides the standard viscous-like damping, dissipation due to eddy currents and domain-wall pinning is considered.

In this paper we study some qualitative properties of the solutions of a nonlinear flux-limited equation arising in the transport of morphogens in biological systems. Questions related to the existence of steady states, the finite speed of propagating fronts or the regularization in the interior of the support are studied from analytical and numerical points of view.

The problem of recovering the initial temperature of a body from discrete temperature measurements made at later times is studied. While this problem has a general formulation, the results of this paper are only given in the simplest setting of a finite (one-dimensional), constant coefficient, linear rod. It is shown that with a judicious placement of a thermometer on this rod, the initial temperature profile of the rod can be completely determined by later time measurements. The paper then studies the number of measurements that are needed to recover the initial profile to a prescribed accuracy and provides an optimal reconstruction algorithm under the assumption that the initial profile is in a Sobolev class.

We study the heat equation on a bounded open set $U\subset {ℝ}^d$ supporting a Borel measure. We obtain asymptotic bounds for the solution and prove the weak parabolic maximum principle. We mainly consider self-similar measures defined by iterated function systems with overlaps. The structures of these measures are in general complicated and intractable. However, for a class of such measures that we call essentially of finite type, important information about the structure of the measures can be obtained. We make use of this information to set up a framework to study the associated heat equations in one dimension. We show that the heat equation can be discretized and the finite element method can be applied to yield a system of linear differential equations. We show that the numerical solutions converge to the actual solution and obtain the rate of convergence. We also study the propagation speed problem.

Being a useful technique for solving differential equation, the study investigated the utility/efficiency of Natural Transform coupled with Adomian Decomposition Method (NADM). For this purpose, one-dimensional (1D) fuzzy fractional-order heat equation was split into two hybrid equations consisting of upper and lower solutions by the fast technique (NADM) together with the idea of level-wise fuzzy number. Accordingly, for the process of the said computation of 1D fuzzy fractional-order heat equation, the techniques of Natural Transform and Adomian Decomposition Method were properly applied. Comparatively, the computational processes were carried out rapidly toward the exact solution. Hence, the method was proved as easy, simple and efficient one. Finally, some examples are practiced that confirmed the results and the utility of the method.

A full proof of the Feynman–Kac-type formula for heat equation on a compact Riemannian manifold is obtained using some ideas originating from the papers of Smolyanov, Truman, Weizsaecker and Wittich.^1-3 In particular, the technique exploited in the paper has some common lines with Chernoff theorem, which is one of the basic points of the approach to the topics undertaken in the above-mentioned papers.

Some families of H-valued vector fields with calculable Lie brackets are given. These provide examples of vector fields on path spaces with a divergence and we show that versions of Bismut type formulae for forms on a compact Riemannian manifold arise as projections of the infinite dimensional theory.

In this paper we present a construction of an infinite dimensional separable Hilbert space associated with a norm induced from the Lévy trace. The space is slightly different from the Cesàro Hilbert space introduced in Ref. 1. The Lévy Laplacian is discussed with a suitable domain which is constructed by a rigging of Fock spaces based on a rigging of Hilbert spaces with the Lévy trace. Then the Lévy Laplacian can be considered as the Gross Laplacian acting on a certain countable Hilbert space. By constructing one-parameter group of operators of which the infinitesimal generator is the Lévy Laplacian, we study the existence and uniqueness of solution of heat equation associated with the Lévy Laplacian. Moreover we give an infinite dimensional stochastic process generated by the Lévy Laplacian.

The main objective of this paper is to investigate an extension of the "Volterra-Gross" Laplacian on nuclear algebra of generalized functions. In so doing, without using the renormalization procedure, this extension provides a continuous nuclear realization of the square white noise Lie algebra obtained by Accardi–Franz–Skeide in Ref. 2. An extended-Gross diffusion process driven by a class of Itô stochastic equations is studied, and solution of the related Poisson equations is derived in terms of a suitable λ-potential.

A parabolic partial differential equation $u_t^{\prime }(t,x)=Lu(t,x)$ is considered, where $L$ is a linear second-order differential operator with time-independent (but dependent on $x$) coefficients. We assume that the spatial coordinate $x$ belongs to a finite- or infinite-dimensional real separable Hilbert space $H$. The aim of the paper is to prove a formula that expresses the solution of the Cauchy problem for this equation in terms of initial condition and coefficients of the operator $L$. Assuming the existence of a strongly continuous resolving semigroup for this equation, we construct a representation of this semigroup using a Feynman formula (i.e. we write it in the form of a limit of a multiple integral over $H$ as the multiplicity of the integral tends to infinity), which gives us a unique solution to the Cauchy problem in the uniform closure of the set of smooth cylindrical functions on $H$. This solution depends continuously on the initial condition. In the case where the coefficient of the first-derivative term in $L$ is zero, we prove that the strongly continuous resolving semigroup indeed exists (which implies the existence of a unique solution to the Cauchy problem in the class mentioned above), and that the solution to the Cauchy problem depends continuously on the coefficients of the equation.

We study the fractional Laplacian ${(-\mathrm{\Delta })}^{\sigma /2}$ on the $n$-dimensional torus ${𝕋}^n$, $n\ge 1$. First, we present a general extension problem that describes any fractional power $L^{\gamma }$, $\gamma >0$, where $L$ is a general nonnegative self-adjoint operator defined in an $L^2$-space. This generalizes to all $\gamma >0$ and to a large class of operators the previous known results by Caffarelli and Silvestre. In particular, it applies to the fractional Laplacian on the torus. The extension problem is used to prove interior and boundary Harnack’s inequalities for ${(-\mathrm{\Delta })}^{\sigma /2}$, when $0<\sigma <2$. We deduce regularity estimates on Hölder, Lipschitz and Zygmund spaces. Finally, we obtain the pointwise integro-differential formula for the operator. Our method is based on the semigroup language approach.

This paper concerns a global controllability problem for heat equations. In the absence of control, the solution to the linear heat system globally exists. While for each initial data, we can find a feedback control acting on an internal subset of the space domain such that the corresponding solution to the system blows up at given time.

We study the heat equation on a half-space with a linear dynamical boundary condition. Our main aim is to show that, if the diffusion coefficient tends to infinity, then the solutions converge (in a suitable sense) to solutions of the Laplace equation with the same dynamical boundary condition.