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A necessary-sufficient condition for the existence or nonexistence of global solutions to the following reaction-diffusion equations { u t = Δ u + ψ ( t ) u p , in R N ×( 0 , t ∗ ) , u ( ⋅ , 0 )= u 0 ≥ 0 , in R N , has not been known and remained as an open problem for a few decades. The purpose of this paper is to resolve this problem completely, even for more general source ψ( t) f( u) as follows: There is a global solution to the equation if and only if ∫ 0 ∞ ψ ( t ) f ( ‖ S ( t ) u 0 ‖ ∞ ) ‖ S ( t ) u 0 ‖ ∞ dt < ∞ for some nonnegative and nontrivial u 0 ∈ C 0 ( R N ) . Here, ( S ( t ) ) t ≥ 0 is the heat semigroup on R N .

The purpose of this paper is to give a necessary and sufficient condition for the existence and non-existence of global solutions of the following semilinear parabolic equations u t = Δ u + ψ ( t ) f ( u ) , in Ω × ( 0 , t ∗ ) , under the Dirichlet boundary condition on a bounded domain. In fact, this has remained as an open problem for a few decades, even for the case f ( u ) = u p . As a matter of fact, we prove: there is no global solution for any initial data if and only if ∫ 0 ∞ ψ ( t ) f ( ‖ S ( t ) u 0 ‖ ∞ ) ‖ S ( t ) u 0 ‖ ∞ dt = ∞ for every nonnegative nontrivial initial data u 0 ∈ C 0 ( Ω ) . Here, ( S ( t ) ) t ≥ 0 is the heat semigroup with the Dirichlet boundary condition.

The purpose of this paper is to give a necessary and sufficient condition for the existence and non-existence of global solutions of the following semilinear parabolic equations \[ u_{t}=\Delta u+\psi(t)f(u),\,\,\mbox{ in }\Omega\times (0,t^{*}), \] under the Dirichlet boundary condition on a bounded domain. In fact, this has remained as an open problem for a few decades, even for the case $f(u)=u^{p}$. As a matter of fact, we prove: \[ \begin{aligned} &\mbox{there is no global solution for any initial data if and only if } &\mbox{the function } f \mbox{ satisfies} &\hspace{20mm}\int_{0}^{\infty}\psi(t)\frac{f\left(\lVert S(t)u_{0}\rVert_{\infty}\right)}{\lVert S(t)u_{0}\rVert_{\infty}}dt=\infty &\mbox{for every }\,\epsilon>0\,\mbox{ and nonnegative nontrivial initial data }\,u_{0}\in C_{0}(\Omega). \end{aligned} \] Here, $(S(t))_{t\geq 0}$ is the heat semigroup with the Dirichlet boundary condition., Comment: 12 pages

In this paper, we study long-time behaviours of solutions to the discrete reaction-diffusion equations u t = Δ ω u + ψ ( t ) f ( u ) with nontrivial and nonnegative initial data, under the mixed bo...

In this paper, we are interested in long time behaviors of solutions to the discrete Laplacian parabolic equations u t = Δ ω u + ψ f ( u ) with nonnegative and non-trivial initial data. In particular, we assume that the function f is convex only on a short interval and f ( α s ) ≈ f ( α ) f ( s ) for 0 α 1 , s > 0 , and we present a critical set depending on the function ψ in the sense that if f is in the critical set, then solutions blow up in finite time for any initial data, and if not, then solutions are global or blow up according to the size of initial data.

In this paper, we study the discrete eigenvalue problems for the p -Schrodinger equations under the mixed boundary conditions defined on networks as follows: − Δ p , ω ϕ ( x ) + q ( x ) | ϕ ( x ) | p − 2 ϕ ( x ) = λ | ϕ ( x ) | p − 2 ϕ ( x ) , x ∈ S , μ ( z ) ∂ ϕ ∂ p n ( z ) + σ ( z ) | ϕ ( z ) | p − 2 ϕ ( z ) = 0 , z ∈ ∂ S , where p > 1 , q is a real-valued function on a network S , and μ , σ are nonnegative functions on the boundary ∂ S of S , with μ ( z ) + σ ( z ) > 0 , z ∈ ∂ S . Next, we use the above result to provide the existence of a positive solution to the discrete Poisson equation − Δ p , ω u ( x ) + q ( x ) | u ( x ) | p − 2 u ( x ) = f ( x ) , x ∈ S , μ ( z ) ∂ u ∂ p n ( z ) + σ ( z ) | u ( z ) | p − 2 u ( z ) = 0 , z ∈ ∂ S .

In this paper, we consider the following discrete p-Laplacian nonlinear parabolic equations with q-nonlocal reaction { u t ( x , t ) = Δ p , ω u ( x , t ) + λ ∑ y ∈ S | u ( y , t ) | q − 1 u ( y , t ) , ( x , t ) ∈ S × ( 0 , ∞ ) , u ( x , t ) = 0 , ( x , t ) ∈ ∂ S × ( 0 , ∞ ) , u ( x , 0 ) = u 0 ≥ 0 , x ∈ S ‾ . Here, S is a network with boundary ∂S. The goal of this paper is to characterize completely the parameters p > 1 and q > 0 to see when the solution blows up, vanishes, or exists globally. Indeed, the blow-up rates for the blow-up solutions are derived.

In this paper, we consider an initial-boundary value problem of the p-Laplacian parabolic equations \begin{equation} \begin{cases} u_{t}\left(x,t\right)=\mbox{div}(|\nabla u\left(x,t\right)|^{p-2}\nabla u(x,t))+f(u(x,t)), & \left(x,t\right)\in \Omega\times\left(0,+\infty\right), \newline u\left(x,t\right)=0, & \left(x,t\right)\in\partial \Omega\times\left[0,+\infty\right), \newline u\left(x,0\right)=u_{0}\geq0, & x\in\overline{\Omega}, \end{cases} \end{equation} where $p\geq2$ and $\Omega$ is a bounded domain of $\mathbb{R}^{N}$ $(N\geq1)$ with smooth boundary $\partial\Omega$. The main contribution of this work is to introduce a new condition \[ \mbox{$(C_{p})$$\hspace{1cm} \alpha \int_{0}^{u}f(s)ds \leq uf(u)+\beta u^{p}+\gamma,\,\,u>0$} \] for some $\alpha, \beta, \gamma>0$ with $0, Comment: 15 pages. arXiv admin note: text overlap with arXiv:1706.03494

In this paper, we discuss and answer the following dichotomy problems: Let S be a network and Δ p , ω be a discrete p -Laplace operator with 1 p ∞ . (i) If u , v are functions satisfying − Δ p , ω u x ≤ − Δ p , ω v x , x ∈ S , u z ≤ v z , z ∈ ∂ S , then either u ≡ v on S ¯ or u v in S . (ii) If u , v are functions satisfying u t x , t − Δ p , ω u x , t ≤ v t x , t − Δ p , ω v x , t , x , t ∈ S × 0 , T , u x , 0 ≤ v x , 0 , x ∈ S , u z , t ≤ v z , t , z , t ∈ ∂ S × 0 , T then either u ≡ v on S ¯ or u v in S × ( 0 , T ) . We believe that this work is not only interesting in itself, but also gives a clue to solve the problems defined on the continuous domain.

This work is concerned with the extinction and positivity properties of solutions to the discrete p-Laplacian parabolic equation u t = Δ p , ω u − f ( u ) , p > 1 with a nonlinear absorption on a weighted graph. We provide a complete characterization for f to see when we have an extinctive solution or a positive solution. In addition, the extinction time is estimated for the extinctive solutions, and the decay rate is given to the positive solutions. Finally, we give several numerical examples which illustrate the main results.

The purpose of this paper is to investigate a condition \begin{equation*} (C_{p}) \hspace{1cm} \alpha \int_{0}^{u}f(s)ds \leq uf(u)+\beta u^{p}+\gamma,\,\,u>0 \end{equation*} for some $\alpha>2$, $\gamma>0$, and $0\leq\beta\leq\frac{\left(\alpha-p\right)\lambda_{p,0}}{p}$, where $p>1$ and $\lambda_{p,0}$ is the first eigenvalue of the discrete $p$-Laplacian $\Delta_{p,\omega}$. Using the above condition, we obtain blow-up solutions to discrete $p$-Laplacian parabolic equations \begin{equation*} \begin{cases} u_{t}\left(x,t\right)=\Delta_{p,\omega}u\left(x,t\right)+f(u(x,t)), & \left(x,t\right)\in S\times\left(0,+\infty\right), \mu(z)\frac{\partial u}{\partial_{p} n}(x,t)+\sigma(z)|u(x,t)|^{p-2}u(x,t)=0, & \left(x,t\right)\in\partial S\times\left[0,+\infty\right), u\left(x,0\right)=u_{0}\geq0(nontrivial), & x\in S, \end{cases} \end{equation*} on a discrete network $S$, where $\frac{\partial u}{\partial_{p}n}$ denotes the discrete $p$-normal derivative. Here, $\mu$ and $\sigma$ are nonnegative functions on the boundary $\partial S$ of $S$, with $\mu(z)+\sigma(z)>0$, $z\in \partial S$. In fact, it will be seen that the condition $(C_{p})$, the generalized version of the condition $(C)$, improves the conditions known so far., Comment: 22 pages. arXiv admin note: substantial text overlap with arXiv:1706.03494

We find all real-valued general solutions$f:S\rightarrow \mathbb{R}$of the d’Alembert functional equation with involution$$\begin{eqnarray}\displaystyle f(x+y)+f(x+\unicode[STIX]{x1D70E}y)=2f(x)f(y) & & \displaystyle \nonumber\end{eqnarray}$$for all$x,y\in S$, where$S$is a commutative semigroup and$\unicode[STIX]{x1D70E}~:~S\rightarrow S$is an involution. Also, we find the Lebesgue measurable solutions$f:\mathbb{R}^{n}\rightarrow \mathbb{R}$of the above functional equation, where$\unicode[STIX]{x1D70E}:\mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$is a Lebesgue measurable involution. As a direct consequence, we obtain the Lebesgue measurable solutions$f:\mathbb{R}^{n}\rightarrow \mathbb{R}$of the classical d’Alembert functional equation$$\begin{eqnarray}\displaystyle f(x+y)+f(x-y)=2f(x)f(y) & & \displaystyle \nonumber\end{eqnarray}$$for all$x,y\in \mathbb{R}^{n}$. We also exhibit the locally bounded solutions$f:\mathbb{R}^{n}\rightarrow \mathbb{R}$of the above equations.

This paper is concerned with long time behaviors of solutions to the initial-boundary value problem of the evolution p -Laplacian equations with nonlinear absorption. A long time behavior of solutions to these equations has been studied so far for particular types of nonlinear absorption term. The purpose of this paper is to give a complete characterization of the nonlinear absorption term, according to the parameter p in p -Laplacian operator and the growth of the nonlinear absorption term near the origin, in order to determine whether the solution to the equations is extinctive or positive. In addition, we also give upper bounds for extinction times of solutions.

In this paper, we discuss the extinction and positivity of solutions to evolution p -Laplacian equations u t - Δ p , ω u + µ | u | q - 1 u = 0 with absorption on finite networks with p > 1 , q > 0 , and µ ? 0 . We present necessary and sufficient conditions for parameters p , q , and µ in order that solutions to the equation are extinctive or positive, respectively. In addition, when a solution is positive, we provide its decay rate and when a solution becomes extinctive in finite time, we give estimate for the extinction time.

In this paper, we discuss the conditions under which blow-up occurs for the solutions of reaction-diffusion equations on networks. The analysis of this class of problems includes the existence of blow-up in finite time and the determination of the blow-up time and the corresponding blow-up rate. In addition, when the solution blows up, we give estimates for the blow-up time and also provide the blow-up rate. Finally, we show some numerical illustrations which describe the main results.

This is an article to introduce discrete nonlinear p-Laplacian parabolic equations on networks and discuss the conditions under which blow-up occurs for the solutions. We first deal with the case p = 2, introducing a recent result about the blow-up phenomena for the solutions. Secondly, we deal with the general p-Laplacian case. In each case, we classify the parameters depending on the equations so that we can see when the solutions blow up or globally exist. Moreover, the blow-up time and blow-up rate are introduced for the blow-up solutions. The last part is devoted to the blow-up of Fujita type.

The purpose of this paper is to introduce a new condition \[ \hbox{(C)$\hspace{1cm} \alpha \int_{0}^{u}f(s)ds \leq uf(u)+\beta u^{2}+\gamma,\,\,u>0$} \] for some $\alpha, \beta, \gamma>0$ with $0, Comment: 19 pages

LetSbe a commutative semigroup if not otherwise specified andf:S→ℝ. In this paper we consider the stability of exponential functional equations|f(x+σ(y))-g(x)f(y)|≤ϕ(x)or ϕ(y),|f(x+σ(y))-f(x)g(y)|≤ϕ(x)orϕ(y)for allx,y∈Sand whereσ:S→Sis an involution. As main results we investigate bounded and unbounded functions satisfying the above inequalities. As consequences of our results we obtain the Ulam-Hyers stability of functional equations (Chung and Chang (in press); Chávez and Sahoo (2011); Houston and Sahoo (2008); Jung and Bae (2003)) and a generalized result of Albert and Baker (1982).

We address some forward and inverse problems involving indefinite eigenvalues for discretep-Laplacian operators with potential terms. These indefinite eigenvalues are the discrete analogues ofp-Laplacians on Riemannian manifolds with potential terms. We first define and discuss some fundamental properties of the indefinite eigenvalue problems for discretep-Laplacian operators with potential terms with respect to some given weight functions. We then discuss resonance problems, anti-minimum principles, and inverse conductivity problems for the discretep-Laplacian operators with potential terms involving the smallest indefinite eigenvalues.

In this paper, we show that for a given convolutor Q for tempered Gevrey distribution and positive number M, a Gevrey distribution T whose Fourier transform is supported in the set {ξ∈ℝd:|ˆ Q(ξ)|≤M} can be characterized by the behaviour of the sequence of the successive convolute powers of Q acting on T. We also deal with the ‘dual’ characterization – a Gevrey distribution T supported in {ξ∈ℝd: |ˆ Q(ξ)|≥M} can be characterized by the sequence , where T0=T, and Q* Tk+1=Tk, k=0, 1, 2, …. As an application, we discuss the problem on sequences characterizing the sine functions and get the most general result of those obtained before.

In this paper, we consider a discrete version of the following p-Laplacian evolution equation u t = Δ p u , with p > 1 , on a network. Using the discrete p-Laplacian operator in graphs which means a nonlinear diffusion phenomenon on networks, we first introduce the p-Laplacian evolution equation on networks. In fact, spatial derivative in p-Laplacian evolution equation, comparing the continuous case, is replaced with the discrete p-Laplacian operator. Thus, the resulting system is semi-discrete, discrete in space and continuous in time. The main concern is to investigate the large time behaviors of nontrivial solutions of this equation whose initial data are nonnegative and the boundary data vanish. In order to do so, we use the analytic approaches such as vector calculus on networks, maximum principle and comparison principle instead of numerical ones. After deriving the basic properties of this equation, we finally prove that the solution becomes extinct for 1 p 2 and remains strictly positive for p ⩾ 2 .

This paper provides an algorithm for identifying the multipoles of a meromorphic function from the boundary measurements, which can be applied to inverse source problems for a Poisson equation to reconstruct its source such as linear combination of mono and dipolar point sources. After establishing an algorithm that allows to identify the number, the locations and the moments of poles, we provide a stability estimate of the algorithm under the measurement error.

In this paper, we deal with the discrete p-Laplacian opera- tors with a potential term having the smallest nonnegative eigenvalue. Such operators are classied as its smallest eigenvalue is positive or zero. We discuss differences between them such as an existence of solutions of p-Laplacian equations on networks and properties of the energy func- tional. Also, we give some examples of Poisson equations which suggest a difference between linear types and nonlinear types. Finally, we study characteristics of the set of a potential those involving operator has the smallest positive eigenvalue.

In this paper, we define the discrete p-Schrodinger operators on finite networks and discuss the existence of the Dirichlet eigenvalues and their eigenfunctions for the operators. We also provide various equivalent conditions for the existence of positive solutions for Dirichlet boundary value problems for the operators.

In this paper, we show the existence of a positive solution of nonlinear discrete boundary value problems for the discrete p -Laplacian with potential term. Moreover, the multiplicity of solutions for the problems is also investigated.

In this paper, we discuss the existence and the uniqueness of discrete boundary value problems involving the p-Laplacian with potential terms. To this end, we deal with the strong comparison principle and the Dirichlet principle for the p-Laplacian with potential terms on weighted graphs with boundary. Moreover, we provide a lower bound of the potential terms which guarantees the existence of solutions of the boundary value problem for the operator. Finally, we also discuss inverse conductivity and potential problems for the operator.

The purpose of this work is to identify the internal structure of the electrical networks with data obtained from only a part of network or the boundary of network. To be precise, it is discussed whether we can identify resistors or electrical conductivities of each link inside networks by the measurement of voltage on the boundary which is induced by a prescribed current on the boundary. As a result, it is shown that the structure of the resistor network can be determined uniquely by only one pair of the data (current, voltage) on the boundary, if the resistors satisfy an appropriate condition. Besides, several useful results about the energy functionals, which means the electrical power, are included.

We develop a comparison principle for the discrete p-Laplacian Δ p,ω as follows: Let S be a connected graph. Assume that f 1 and f 2 are functions on S satisfying If f 1≢f 2 on S, then f 1

In this paper, we characterize Sobolev spaces Hs(Rn), s ∈ R by the initial value of solutions of heat equation with a growth condition. By using an idea in its proof, we also discuss a stability problem for Cauchy functional equation in the Sobolev spaces.

In this paper, we consider the general solution of quadratic functional equation f(ax+y)+f(ax−y)=f(x+y)+f(x−y)+2(a2−1)f(x) for any integer a with a≠−1,0,1. Moreover we reformulate and prove the Hyers–Ulam–Rassias stability theorem of the above equation in the spaces of tempered distributions and Fourier hyperfunctions. The generalized Hyers–Ulam stability originated from the Th.M. Rassias's stability theorem that appeared in his paper [Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978) 297–300].

Let E(x, ξ, t) be the Mehler kernel and let U(x, t) be the solution, with some appropriate growth estimate (*), of the Hermite heat equation in R×(0, T). Then there exists a unique Fourier hyperfunction u such that U(x, t)=⟨ u(ξ), E(x, ξ, t)⟩. Conversely for any Fourier hyperfunction u, U(x, t):=⟨ u(ξ), E(x, ξ, t)⟩ is the smooth solution of the Hermite heat equation in R×(0, T) satisfying the growth estimate (*).

In this paper, we discuss discrete versions of the heat equations and the wave equations, which are called the ω-diffusion equations and the ω-elastic equations on graphs. After deriving some basic properties, we solve the ω-diffusion equations under (i) the condition that there is no boundary, (ii) the initial condition and (iii) the Dirichlet boundary condition. We also give some additional interesting properties on the ω-diffusion equations, such as the minimum and maximum principles, Huygens property and uniqueness via energy methods. Analogues of the ω-elastic equations on graphs are also discussed.

Suppose a spherical conductor has a single dipole. If it is assumed that the direction of current is perpendicular to the location of dipole, then we prove that the dipole can be obtained as an exact form from the induced magnetic field and provide an easy algorithm to locate the dipole from the samples of a magnetic field outside the sphere without using any series approximation. A numerical implementation is also presented. Copyright © 2006 John Wiley & Sons, Ltd.