N-Laplacian and N/2-Hessian Type Equations with Exponential Reaction Terms and Measure Data

SummaryWe analyze both a priori and a posteriori error analysis of finite‐element method for elliptic optimal control problems with measure data in a bounded convex domain in (d = 2or3). The solution of the state equation of such type of problems exhibits low regularity due to the presence of measure data, which introduces some difficulties for both theory and numerics of the finite‐element method. We first prove the existence, uniqueness, and regularity of the solution to the optimal control problem. To discretize the control problem, we use continuous piecewise linear elements for the approximations of the state and co‐state variables, whereas piecewise constant functions are used for the control variable. We derive a priori error estimates of order for the state, co‐state, and control variables in the L2‐norm. Further, global a posteriori upper bounds for the state, co‐state, and control variables in the L2‐norm are established. Moreover, local lower bounds for the errors in the state and co‐state variables and a global lower bound for the error in the control variable are obtained in the case of two space dimensions (d = 2). Numerical experiments are provided, which support our theoretical results.

The study of the boundary value problem with arbitrary measurable data originated in the dissertation of Luzin where he investigated the Dirichlet problem for harmonic functions in the unit disk. Recently, in \cite{R7}, we studied the Hilbert, Poincaré and Neumann boundary value problems with arbitrary measurable data for generalized analytic and generalized harmonic functions and provided applications to relevant problems in mathematical physics. The present paper is devoted to the study of the boundary value problem with arbitrary measurable boundary data in a domain with rectifiable boundary corresponding to semi-linear equation with suitable nonlinear source. We construct a completely continuous operator and generate nonclassical solutions to the Hilbert and Poincaré boundary value problems with arbitrary measurable data for Vekua type and Poisson equations, respectively. Based on that, we prove the existence of solutions of the Hilbert boundary value problem for the nonlinear Vekua type equation with arbitrary measurable data in a domain with rectifiable boundary. It is necessary to point out that our approach differs from the classical variational approach in PDE as it is based on the geometric interpretation of boundary values as angular (along non-tangential paths) limits. The latter makes it possible to also obtain a theorem on the boundary value problem for directional derivatives, and, in particular, of the Neumann problem with arbitrary measurable data for the Poisson equation with nonlinear sources in any Jordan domain with rectifiable boundary. As a result we arrive at applications to some problems of mathematical physics.

Spontaneous Signal Generation in Living Cells

This paper is concerned with the study of damped wave equation of Kirchhoff type in , with initial and Dirichlet boundary condition, where Ω is a bounded domain of having a smooth boundary ∂ Ω. Under the assumption that g is a function with exponential growth at infinity, we prove global existence and the decay property as well as blow-up of solutions in finite time under suitable conditions. MSC: 35L70, 35B40, 35B44.

Continuity estimates for porous medium type equations with measure data

A floating structure has many options for effective ocean space utilization, for instance, the well known floating airport project, called Mega-Float. But after the end of the project, small scale floating structure began to be paid to attention. As the good example of such a kind of floating structure, there is the floating restaurant named “WATERLINE” (Figure 1) in Tokyo Bay. “WATERLINE” is small scale floating structure, and it is moored at the Tennoz Canal that is the closed water area. Therefore, when the ship passes around the floating restaurant, ship wave forces give a great influence on dynamic behavior of floating restaurant. As for ship waves, several studies have been made on the wave resistance and influence on ship handling concerning ship waves, but it is hardly to find papers focused on influence that ship wave forces give to dynamic behavior of small scale floating structure. In this research, dynamic behavior of small scale floating structure by ship wave forces was studied through both theoretical and experimental approach. As for the theoretical analysis, the equations of the Boussinesq type to treat shallow water area were adopted, and ADI (Alternating Direction Implicit) method in a numerical calculation was used for the analysis of these equations. And the floating structure was assumed to be a rigid body, and the displacement responses by ship wave forces were analyzed. With regard to experimental study, dynamic behavior of “WATERLINE” and wave height by the ship wave were actually measured. This measurement data is a profitable basic data for other researchers and engineers in order to analyze a floating structure. In the present paper, the validity of the numerical calculation program for ship wave response analysis was verified by the comparison between calculation results and the measurement results, the characteristics of the displacement response and the wave height were discussed by the numerical results that had been obtained by changing by the ship’s speed and the distance between floating structure and the ship. In addition, the evaluation of habitability in vertical motion of the small scale floating structure at the canal was examined by the diagram proposed from our research results [1], [2]. And, in regard to the ship that passes over around floating structure, ship’s speed limit and minimum distance between the ship and the floating structure were proposed.

We consider the inhomogeneous porous medium equation $$\partial_t u - \Delta u^m = \mu, \qquad m>\frac{(N-2)_+}{N},$$ and more general equations of porous medium type with a non-negative Radon measure μ on the right-hand side. In a first step, we prove a priori estimates for weak solutions in terms of a linear Riesz potential of the right-hand side measure, which takes exactly the same form as the one for the classical heat equation. Then, we give an optimal criterium for the continuity of weak solutions, again in terms of a Riesz potential. Finally, we prove the existence of non-negative very weak solutions, and show that these constructed very weak solutions satisfy the same estimates. We deal with both the degenerate case \(m>1\), and the singular case \(\frac{(N-2)_+}{N}<m<1\).KeywordsWeak SolutionPotential EstimateNonlinear Parabolic EquationPorous Medium EquationRiesz PotentialThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Let Ω be a bounded domain of ℝN, and Q = Ω × (0, T). We consider problems of the type [Formula: see text] where Δpis the p-Laplacian, μ is a bounded Radon measure, u0∈ L1(Ω), and ±𝒢(u) is an absorption or a source term. In the model case 𝒢(u) = ±|u|q-1u (q &gt; p-1), or 𝒢, has an exponential type. We prove the existence of renormalized solutions for any measure μ in the subcritical case, and give sufficient conditions for existence in the general case, when μ is good in time and satisfies suitable capacitary conditions.

We consider obstacle problems with measure data related to elliptic equations of p-Laplace type, and investigate the connections between low order regularity properties of the solutions and non-linear potentials of the data. In particular, we give pointwise estimates for the solutions in terms of Wol potentials and address the questions of boundedness and continuity of the solution.

In this paper, a novel mathematical model based on RC circuit for investigating the gel battery is presented. The proposed mathematical model consists of a polynomial term and an exponential term which can present the discharging behavior of a battery. The polynomial term represents the stable discharging behavior and the exponential term represents the unstable discharging behavior. The proposed model proved that the aging of a battery increases with cycle numbers and reflects the variation in the parameters of the model in state of charge (SoC) of a battery. In addition, a simpler algorithm that estimates part of the discharging characteristics to predict SoC of a battery also presented in this work. The algorithm has shorter computing time, higher efficiency and greater convenience. Through the experiments that practical operations, maximum absolute percentage error of SoC for the relationship between measurement and fitting data is 1.83%. The results mean that the proposed model has been verified, demonstrating its effectiveness and accuracy.

General methods of successive approximations to calculate the relaxation spectrum from data of dynamic mechanical measurements are worked out using an inversion theorem of integral equations of the Stieltjes type. The idea underlying the methods is essentially the same as that used by Schwarzl and Staverman in their treatment of stress relaxation data. It is shown that the delta-function method of Schwarzl and Staverman and of Leaderman is not a unique process of successive approximations to obtain relaxation spectra from dynamic data. Conditions to obtain a series of best approximations in terms of the delta-function method are discussed.

We discuss the algebro-geometric initial value problem for the Ablowitz-Ladik hierarchy with complex-valued initial data and prove unique solvability globally in time for a set of initial (Dirichlet divisor) data of full measure. To this effect we develop a new algorithm for constructing stationary complex-valued algebro-geometric solutions of the Ablowitz-Ladik hierarchy, which is of independent interest as it solves the inverse algebro-geometric spectral problem for general (non-unitary) Ablowitz-Ladik Lax operators, starting from a suitably chosen set of initial divisors of full measure. Combined with an appropriate first-order system of differential equations with respect to time (a substitute for the well-known Dubrovin-type equations), this yields the construction of global algebro-geometric solutions of the time-dependent Ablowitz-Ladik hierarchy. The treatment of general (non-unitary) Lax operators associated with general coefficients for the Ablowitz-Ladik hierarchy poses a variety of difficulties that, to the best of our knowledge, are successfully overcome here for the first time. Our approach is not confined to the Ablowitz-Ladik hierarchy but applies generally to (1+1)-dimensional completely integrable soliton equations of differential-difference type.

We discuss the algebro-geometric initial value problem for the Ablowitz-Ladik hierarchywith complex-valued initial data and prove unique solvability globally intime for a set of initial (Dirichlet divisor) data of full measure. To this effect we develop a new algorithm for constructing stationary complex-valued algebro-geometric solutions of the Ablowitz-Ladik hierarchy, which is of independent interest as it solves the inversealgebro-geometric spectral problem for general (non-unitary) Ablowitz-Ladik Lax operators, starting from a suitably chosen set of initial divisors of full measure. Combined with an appropriate first-order system of differential equations with respect to time (a substitute for the well-known Dubrovin-type equations), this yields the construction of global algebro-geometric solutions of the time-dependent Ablowitz-Ladik hierarchy. The treatment of general (non-unitary) Lax operators associated with general coefficients for the Ablowitz-Ladik hierarchy poses a variety of difficulties that, to the best of our knowledge, are successfully overcome here for the first time. Our approach is not confined to the Ablowitz-Ladik hierarchy but applies generally to $(1+1)$-dimensional completely integrable soliton equations of differential-difference type.

This paper is concerned with generalized, discontinuous solutions of initial value problems for nonlinear first order partial differential equations. “Layering” is a method of approximating an arbitrary generalized solution by dividing its domain, say a half-space t≥0, into thin layers (i-ℓ)h≤t≤ih, i=1,2,...(h&gt;0), and using a strict solution u i in the i-th layer. On the interface t=(i-ℓ)h, u t is required to reduce to a smooth function approximating the values on that plane of u i-ℓ . The resulting stratified configuration of strict solutions of the equation is called a “layered solution” . Under appropriate conditions, any generalized solution can be realized as the limit of a sequence of layered solutions for which smoothing is made finer and finer and h→0; the estimates needed to prove this pertain solely to strict solutions of the equation concerned. Layering was first used by N.N. Kuznetsov in connection with conservation laws and with initial data of bounded variation (in a multi-dimensional sense). These matters are also discussed here, the method extended to the case of bounded, measurable initial data, and a large class of possible smoothing operations discussed. In addition, the method is adapted to equations of Hamilton-Jacobi type.

In the present article we study global existence for a nonlinear parabolic equation having a reaction term and a Radon measure datum: where 1 &lt; p &lt; N, Ω is a bounded open subset of ℝN (N ≥ 2), Δpu = div(|∇u|p−2∇u) is the so called p-Laplacian operator, sign s ., ϕ(ν0) ∈ L1(Ω), μ is a finite Radon measure and f ∈ L∞(Ω×(0, T)) for every T &gt; 0. Then we apply this existence result to show wild nonuniqueness for a connected nonlinear parabolic problem having a gradient term with natural growth.