Global Existence Result Research Articles

Long-time Existence for Systems of Quasilinear Wave Equations

We consider quasilinear wave equations in (1 + 3)-dimensions where the nonlinearity $$F(u, u^{\prime } , u^{\prime \prime })$$ is permitted to depend on the solution rather than just its derivatives. For scalar equations, if $$(\partial _{u}^{2}F)(0, 0, 0) = 0$$ , almost global existence was established by Lindblad. We seek to show a related almost global existence result for coupled systems of such equations. To do so, we will rely upon a variant of the $$r^p$$ -weighted local energy estimate of Dafermos and Rodnianski that includes a ghost weight akin to those used by Alinhac. The decay that is needed to close the argument comes from space–time Klainerman–Sobolev type estimates from the work of Metcalfe, Tataru, and Tohaneanu.

Existence of global solutions and blow-up results for a class of p(x)−Laplacian heat equations with logarithmic nonlinearity

This paper?s main objective is to examine an initial boundary value problem of a quasilinear parabolic equation of non-standard growth and logarithmic nonlinearity by utilizing the logarithmic Sobolev inequality and potential well method. Results of global existence, estimates of polynomial decay, and blowing up of weak solutions have been obtained under certain conditions that will be stated later. Our results extend those of a recent paper that appeared in the literature.

Ramification of Volterra-type rough paths

We extend the new approach introduced in [24] and [25] for dealing with stochastic Volterra equations using the ideas of Rough Path theory and prove global existence and uniqueness results. The main idea of this approach is simple: Instead of the iterated integrals of a path comprising the data necessary to solve any equation driven by that path, now iterated integral convolutions with the Volterra kernel comprise said data. This leads to the corresponding abstract objects called Volterra-type Rough Paths, as well as the notion of the convolution product, an extension of the natural tensor product used in Rough Path Theory.

Global existence and nonexistence of solutions for semilinear wave equation with a new condition

In this paper, we consider the initial-boundary problem for semilinear wave equation with a new condition on the source term $ f(u) $ such as \begin{document}$ \begin{equation*} \alpha \int_0^{u } f(s)ds \leq uf(u) + \beta u^2 +\alpha \sigma, \end{equation*} $\end{document} for some positive constants $ \alpha $, $ \beta $, and $ \sigma $, where $ \alpha >2 $ and $ \beta < \frac{\lambda_1(\alpha -2)}{2} $ with $ \lambda_1 $ being a first eigenvalue of Dirichlet-Laplacian. The condition on $ f(u) $ with $ \beta = 0 $ and $ \sigma = 0 $ was studied in [13] and [3]. By introducing a family of potential wells, we are able to establish the invariant sets, vacuum isolation of solutions, and a threshold result of global existence and nonexistence of solutions with the new condition on a source term. Moreover, we discuss the global existence and nonexistence of solutions for problem with initial condition $ E(0)<d $ and critical initial condition $ E(0) = d $.

On the p-Laplacian type equation with logarithmic nonlinearity: existence, decay and blow up

This work is deal with a problem of wave equation with p-Laplacian, strong damping and logarithmic source terms under initial-boundary conditions. The global existence of weak solution was proved for related to the equation. Global existence results of solutions are obtained using the potential well method, Galerkin method and compactness approach corresponding to the logarithmic source term. Besides, we established the energy functional decaying polynomially to zero as the time goes to infinity due to Nakao?s inequality and some precise priori estimates on logarithmic nonlinearity. For suitable conditions we proved the finite time blow up results of solutions. The proof is based on the concavity method, perturbation energy method and differential-integral inequality technique. Additionally, under suitable assumptions on initial data, the infinite time blow up result is investigated with negative initial energy.

Conformal-type energy estimates on hyperboloids and the wave-Klein-Gordon model of self-gravitating massive fields

<abstract><p>In this article we revisit the global existence result of the wave-Klein-Gordon model of the system of the self-gravitating massive field. Our new observation is that, by applying the conformal energy estimates on hyperboloids, we obtain mildly increasing energy estimate up to the top order for the Klein-Gordon component, which clarify the question on the hierarchy of the energy bounds of the Klein-Gordon component in our previous work. Furthermore, a uniform-in-time energy estimate is established for the wave component up to the top order, as well as a scattering result. These improvements indicate that the partial conformal symmetry of the Einstein-massive scalar system will play an important role in the global analysis.</p></abstract>

Examples of finite time blow up in mass dissipative reaction-diffusion systems with superquadratic growth

<p style='text-indent:20px;'>We provide explicit examples of finite time <inline-formula><tex-math id="M1">\begin{document}$ L^\infty $\end{document}</tex-math></inline-formula>-blow up for the solutions of <inline-formula><tex-math id="M2">\begin{document}$ 2\times 2 $\end{document}</tex-math></inline-formula> reaction-diffusion systems for which three main properties hold: positivity is preserved for all time, the total mass is uniformly controlled and the growth of the nonlinear reaction terms is superquadratic. They are obtained by choosing the space dimension large enough. This is to be compared with recent global existence results of uniformly bounded solutions for the same kind of systems with quadratic or even slightly superquadratic growth depending on the dimension. Such blow up may occur even with homogeneous Neumann boundary conditions. All these <inline-formula><tex-math id="M3">\begin{document}$ L^\infty $\end{document}</tex-math></inline-formula>-blowing up solutions may be extended as weak global solutions. Blow up examples are also provided in space dimensions one, two and three with various growths.</p>

Traveling waves for a nonlinear Schrödinger system with quadratic interaction

We study traveling wave solutions for a nonlinear Schr\"odinger system with quadratic interaction. For the non mass resonance case, the system has no Galilean symmetry, which is of particular interest in this paper. We construct traveling wave solutions by variational methods and see that for the non mass resonance case there exist specific traveling wave solutions which correspond to the solutions for ``zero mass" case in nonlinear elliptic equations. We also establish the new global existence result for oscillating data as an application. Both of our results essentially come from the lack of Galilean invariance in the system.

Local existence, global existence and decay results of a logarithmic wave equation with delay term

In this article, we deal with a logarithmic wave equation with strong damping and delay. Firstly, we prove the local existence by utilizing the semigroup theory. Later, we obtain the global existence of solutions by using the well‐depth method. Moreover, under appropriate assumptions on the weight of the delay and that of strong damping, we get the exponential decay.

Global existence for the quadratic Dirac equation in two and three space dimensions

In this paper we study global existence and pointwise decay estimates for the nonlinear Dirac equation with quadratic nonlinearity. We consider four cases depending on the spatial dimension n, the mass parameter m, and the initial data ψ0: i) (n,m)=(2,0) and ψ0 is compactly supported; ii)(n,m)=(3,0) and ψ0 is compactly supported; iii)(n,m)=(3,0) and ψ0 is not necessarily compactly supported; iv)n=3, m≥0 and ψ0 is compactly supported. In each of the cases i)-iii), we prove a small data global existence result, a sharp pointwise decay estimate and a scattering result for the global solution. In the case iv) we prove a uniform (in the mass parameter m) global existence result, a unified pointwise decay estimate, and a scattering result.

Weak solutions to triangular cross diffusion systems modeling chemotaxis with local sensing

New estimates and global existence results are provided for a class of systems of cross diffusion equations arising from the modeling of chemotaxis with local sensing, possibly featuring a growth term of logistic-type as well. For sublinear non-increasing motility functions, convergence to the spatially homogeneous steady state is shown, a dedicated Lyapunov functional being constructed for that purpose.

On a Predator-Prey Model Involving Age and Spatial Structure

In this paper, we study the mathematical analysis of a nonlinear age-dependent predator–prey system with diffusion in a bounded domain with a non-standard functional response. Using the fixed point theorem, we first show a global existence result for the problem with spatial variable. Other results of existence concerning the spatial homogeneous problem and the stationary system are discussed. At last, numerical simulations are performed by using finite difference method to validate the results.

Global classical solvability in a food metric chemotaxis system under zero boundary conditions at infinity

We consider the nutrient–chemotaxis model, derived by a food metric, on the real line. The model consists of the equation for the nutrient density of ordinary differential equation type and that for the microorganism density of parabolic type, in which the diffusion coefficient is singular at the zero nutrient density. We establish a global existence result of bounded classical solutions for a large class of initial data under zero boundary conditions at infinity. To this end, we propose a transformation related to the food metric that transforms the original system into a parabolic–hyperbolic system with linear diffusion. We obtain a proper global well-posedness result for the transformed system using standard estimates, including the heat kernel estimates. We convert this to the solution of the original equation via an inverse transformation.

On the nonlinear Schrödinger equation in spaces of infinite mass and low regularity

We study the nonlinear Schrödinger equation with initial data in $ \mathcal Z ^s_p(\mathbb R^d)=\dot{H}^s(\mathbb R^d)\cap L^p(\mathbb R^d)$, where $0 < s < \min\{d/2,1\}$ and $2 < p < 2d/(d-2s)$. After showing that the linear Schrödinger group is well-defined in this space, we prove local well-posedness in the whole range of parameters $s$ and $p$. The precise properties of the solution depend on the relation between the power of the nonlinearity and the integrability $p$. Finally, we present a global existence result for the defocusing cubic equation in dimension three for initial data with infinite mass and energy, using a variant of the Fourier truncation method.

Global existence results for solutions of general conservative advection-diffusion equations in [formula omitted

In this paper a rigorous study concerning global existence results and estimates for the sup norm of non-negative bounded solutions for one-dimensional advection-diffusion equations ut+(b(x,t)uk+1)x=μ(t)uxx, with 0≤k<2 and initial data u0∈L1(R)∩L∞(R) is provided using a technique based on energy methods. In respect of the arbitrary advective speed term, it is only assumed that b(x,t) and ∂b(x,t)/∂x are limited.

Existence and finite time blow-up for nonlinear Schrödinger equations in the Bopp-Podolsky electrodynamics

In this paper, we consider the nonlinear Schrödinger equations in the Bopp-Podolsky electrodynamics. By using the fixed point theorem and the method of Strichartz's estimate, we construct local solutions in some spatial-time space and further prove global existence result for small data. Moreover, we establish the virial identity for the system and show that solutions with negative energy blow up in finite time.

The zero mass problem for Klein–Gordon equations

In this paper, we are interested in the global existence result for a class of Klein–Gordon equations, particularly in the unified time decay result concerning a possibly vanishing mass parameter. We give for the first time a rigorous proof for this problem, which relies on both the flat foliation and the hyperboloidal foliation of the Minkowski spacetime. To take advantage of both foliations, an iteration procedure is used.

The Cauchy problem and continuation of periodic solutions for a generalized Camassa–Holm equation

We consider a three-parameter family of non-linear equations with -order non-linearities. Such a family includes as a particular member the well-known b-equation, which encloses the famous Camassa–Holm equation. For certain choices of the parameters, we establish a global existence result and show a scenario that prevents the wave breaking of solutions. Also, we explore unique continuation properties for some values of the parameters.

Weak shock wave interactions in isentropic Cargo‐LeRoux model of flux perturbation

In this paper, we explore the weak shock wave interactions in isentropic Cargo‐LeRoux model of flux perturbation which describes strictly hyperbolic quasilinear system of conservation laws. For the Riemann problem, we provide a global existence and uniqueness result of exact solution. Further, we establish the exact solution explicitly in terms of one parameter family of elementary wave curves. We devise a necessary and sufficient condition on initial data for which the solution of the Riemann problem consists of either a shock wave or a simple wave according to one and three family of characteristic curves. Lastly, we analyze the Von Neumann result related to collision of weak shock waves of same family.

A Global Existence Result for a Semilinear Wave Equation with Lower Order Terms on Compact Lie Groups

In this paper, we study the semilinear wave equation with lower order terms (damping and mass) and with power type nonlinearity \(|u|^p\) on compact Lie groups. We will prove the global in time existence of small data solutions in the evolution energy space without requiring any lower bounds for \(p>1\). In our approach, we employ some results from Fourier analysis on compact Lie groups.