Arbitrary Initial Energy Research Articles

Blow Up of Solutions to Wave Equations with Combined Logarithmic and Power-Type Nonlinearities

In this paper, we study the initial boundary value problem for wave equations with combined logarithmic and power-type nonlinearities. For arbitrary initial energy, we prove a necessary and sufficient condition for blow up at infinity of the global weak solutions. In addition, we derive a growth estimate for the blowing up global solutions.

Blow-Up Phenomena for a Non-Newton Filtration Equation with Local Linear Boundary Dissipation

In this article, we consider the finite time blow-up phenomenon for a class of non-Newton filtration equations with local linear boundary dissipation. Using a modified concavity method, the sufficient condition for the finite time blow-up is given. Furthermore, we obtain the result of the blow-up solution with arbitrary high initial energy. Meanwhile, the upper and lower estimates of the blow-up time are discussed.

Infinite time blow‐up with arbitrary initial energy for a damped plate equation

AbstractThis paper deals with the infinite blow‐up phenomena for a class of damped plate equations with logarithmic nonlinearity under the Navier boundary condition. Combining potential well method and modified differential inequality technique, we establish the infinite blow‐up result of solutions with arbitrary initial energy. In particular, it is not necessary to suppose that the initial velocity and the initial displacement should have the same sign in the sense of the L2 inner product, that is, the solution may blow up at infinity even , more precisely, .

On a singular epitaxial thin‐film growth equation involving logarithmic nonlinearity

This paper is concerned with the well‐posedness and asymptotic behavior for a singular epitaxial thin‐film growth equation with logarithmic nonlinearity under the Navier boundary condition. Based on the technique of cut‐off and combining with Hardy–Sobolev inequality, the technique of Faedo–Galerkin, and multiplier, we establish the local solvability. Meantime, by virtue of the family of potential wells, we obtain the threshold between the existence and nonexistence of the global solution (including the critical case) and give the upper bound of lifespan and the estimate of blow‐up rate. Furthermore, the results of blow‐up with arbitrary initial energy and the lifespan are derived.

On a singular parabolic $ p $-Laplacian equation with logarithmic nonlinearity

<p>In this paper, we considered a singular parabolic $ p $-Laplacian equation with logarithmic nonlinearity in a bounded domain with homogeneous Dirichlet boundary conditions. We established the local solvability by the technique of cut-off combining with the method of Faedo-Galerkin approximation. Based on the potential well method and Hardy-Sobolev inequality, the global existence of solutions was derived. In addition, we obtained the results of the decay. The blow-up phenomenon of solutions with different indicator ranges was also given. Moreover, we discussed the blow-up of solutions with arbitrary initial energy and the conditions of extinction.</p>

Some qualitative properties of weak solution for pseudo‐parabolic equation with viscoelastic term and Robin boundary conditions

AbstractIn this paper, we consider the initial boundary value problem of the generalized pseudo‐parabolic equation containing viscoelastic terms and associated with Robin conditions. We establish first the local existence of solutions by the standard Galerkin method. Then, we prove blow‐up results for solutions when the initial energy is negative or nonnegative but small enough or positive arbitrary high initial energy, respectively. We also establish the lifespan and the blow‐up rate for the weak solution by finding the upper bound and the lower bound for the blow‐up times and the upper bound and the lower bound for the blow‐up rate. For negative energy, we introduce a new method to prove blow‐up results with a sharper estimate for the upper bound for the blow‐up times. Finally, we prove both the global existence of the solution and the general decay of the energy functions under some restrictions on the initial data.

Blow-Up of Solutions to Von Karman Equations with Arbitrary Initial Energy

Blow-Up of Solutions to Von Karman Equations with Arbitrary Initial Energy

Motions of a dimer on a periodic potential.

A dimer on a periodic potential is a simple system that exhibits a surprisingly rich dynamics. This system is conservative, but it is nonlinear and nonintegrable. In a previous work, we evidenced the autoparametric excitation of the relative motion by the center of mass in two limiting cases (very small or very large initial energy, compared to the external potential depth). We extend these results for arbitrary initial energy. The relevant control parameters are the dimer initial energy and the stiffness of the link between the two particles. In this parameter plane, we build a behavior map which classifies the available dynamical regimes of the dimer. The parameters plane can be separated into domains in which the dimer particles are either trapped in adjacent potential wells, slide along the potential, or exhibit more complex motions in which the particles jumps to farthest well or in which the center-of-mass motion is neither monotonous nor periodic. We discuss the thresholds between these domains.

Global well‐posedness and finite time blow‐up for a class of wave equation involving fractionalp‐Laplacian with logarithmic nonlinearity

AbstractIn this paper, we consider the following class of wave equation involving fractionalp‐Laplacian with logarithmic nonlinearitywhere is a bounded domain with Lipschitz boundary, , , and is the critical exponent in the Sobolev inequality. First, via the Galerkin approximations, the existence of local solutions are obtained when . Next, by combining the potential well theory with the Nehari manifold, we establish the existence of global solutions when . Then, via the Pohozaev manifold, the existence of global solutions are obtained when . By virtue of a differential inequality technique, we prove that the local solutions blow‐up in finite time with arbitrary negative initial energy and suitable initial values. Moreover, we discuss the asymptotic behavior of solutions as time tends to infinity. Here, we point out that the main difficulty is the lack of logarithmic Sobolev inequality concerning fractionalp‐Laplacian.

On a singular parabolic p-biharmonic equation with logarithmic nonlinearity

This paper is concerned with the well-posedness and asymptotic behavior of Dirichlet initial boundary value problem for a singular parabolic p-biharmonic equation with logarithmic nonlinearity. We establish the local solvability by the technique of cut-off combining with the methods of Faedo–Galerkin approximation and multiplier. Meantime, by virtue of the family of potential wells, we use the technique of modified differential inequality and improved logarithmic Sobolev inequality to obtain the global solvability, infinite and finite time blow-up phenomena, and derive the upper bound of blow-up time as well as the estimate of blow-up rate. Furthermore, the results of blow-up with arbitrary initial energy and extinction phenomena are presented.

General decay and blow-up results for a class of nonlinear pseudo-parabolic equations with viscoelastic term

In this paper, we consider initial boundary value problem of the generalized pseudo-parabolic equation contain viscoelastic term. We establish firstly the local existence of solutions by Banach fixed point theorem. Then we prove blow-up results for solutions when the initial energy is negative or nonnegative but small enough or positive arbitrary high initial energy, respectively. We also establish the lifespan and the blow-up rate for the weak solution via finding the upper bound and the lower bound for the blow-up times and the upper bound and the lower bound for the blow-up rate. For negative energy, we introduce a new method to prove blow-up results with sharper estimate for upper bound for the blow-up times. Finally, we prove the global existence of the solution and a general decay estimate under some suitable assumptions.

Well‐posedness of solutions for a class of quasilinear wave equations with strong damping and logarithmic nonlinearity

AbstractThis paper investigates the well‐posedness of solutions for the following quasilinear wave equation with strong damping and logarithmic nonlinearity in a bounded domain with homogeneous Dirichlet boundary: , where . By virtue of the classical Faedo–Galerkin method and some technical efforts, we first establish the local well‐posedness of solutions. Then we discuss the dynamical behaviors of solutions in detail: When and , we show that the solutions exist globally with subcritical and critical initial energy, where denotes the Nehari functional with the initial value u0. Especially, under further suitable assumptions about the initial data, we show that the energy functional decays exponentially. When and , we show that the solutions blow up in finite time with subcritical and critical initial energy. Moreover, by removing the restriction , we prove that the solutions may blow up in finite time with arbitrary high initial energy. In particular, we derive the upper and lower bounds of the blow‐up time. When and , we show that the maximal existence time of solutions can be extended to infinity and the solutions blow up at infinity with subcritical, critical, and arbitrary high initial energy.

Blow-up problems for a parabolic equation coupled with superlinear source and local linear boundary dissipation

In this paper, we consider the finite time blow-up results for a parabolic equation coupled with superlinear source term and local linear boundary dissipation. Using a concavity argument, we derive the sufficient conditions for the solutions to blow up in finite time. In particular, we obtain the existence of finite time blow-up solutions with arbitrary high initial energy. We also derive the upper bound and lower bound of the blow up time.

Well-posedness of solutions for the dissipative Boussinesq equation with logarithmic nonlinearity

In this paper, we consider a class of Boussinesq-type equations with logarithmic nonlinearity. By employing the classical Faedo–Galerkin method, we first establish the local well-posedness of solutions. Then we investigate the dynamical behaviors of solutions. More precisely, for the solutions with subcritical or critical initial energy, we prove that they exist globally and are uniformly bounded when I(u0)>0, where I(u0) denotes the Nehari functional with the initial value u0. Moreover, under further appropriate assumptions about the initial data, we derive the exponential energy decay estimates of global solutions. In particular, for the solutions with subcritical or critical initial energy, we show that they can be extended over time (the whole half line) and then blow up at infinite time when I(u0)<0. Last but not least, by developing some new methods, we prove the existence of infinite time blow-up solutions with arbitrary high initial energy.

Blow-up of weak solutions for a porous elastic system with nonlinear damping and source terms

This paper deals with blow-up solutions of the initial boundary value problems for a porous elastic system with nonlinear damping and source terms at arbitrary initial energy level. We estimate the lower bound and upper bound of the lifespan of the blow-up solution, and also estimate the blow-up rate by considering both linear and nonlinear weak damping terms.

Blow up of solutions for a r(x)-Laplacian Lam'{e} equation with variable-exponent nonlinearities and arbitrary initial energy level

In this paper, we consider the nonlinear $r(x)-$Laplacian Lam'{e} equation $$ u_{tt}-Delta_{e}u-divbig(|nabla u|^{r(x)-2}nabla ubig)+|u_{t}|^{m(x)-2}u_{t}=|u|^{p(x)-2}u $$ in a smoothly bounded domain $Omegasubseteq R^{n}, ngeq1$, where $r(.), m(.)$ and $p(.)$ are continuous and measurable functions. Under suitable conditions on variable exponents and initial data, the blow-up of solutions is proved with negative initial energy as well as positive.

Infinite Time Blow-Up of Solutions for A Class of Logarithmic Wave Equations with Arbitrary High Initial Energy

Infinite Time Blow-Up of Solutions for A Class of Logarithmic Wave Equations with Arbitrary High Initial Energy

Arbitrary initial energy blow up for fourth-order viscous damped wave equation with exponential-type growth nonlinearity

This letter is concerned with the following damped nonlinear wave equation with exponential nonlinearity utt+Δ2u+u+Δ2ut+ut=f(u)inΩ×(0,∞)where the nonlinearity f is a regular function satisfying f(0)=0 with an exponential growth to fix later. The above wave problem describes a class of essential nonlinear evolution equations appearing in the elastic–plastic-microstructure models. In this letter blow up result is established for arbitrary initial energy with the viscous damping terms.

Infinite time blow‐up of solutions to a class of wave equations with weak and strong damping terms and logarithmic nonlinearity

AbstractThis paper investigates the infinite time blow‐up of solutions with arbitrary high initial energy to wave equations with weak damping term, strong damping term, and logarithmic nonlinearity. This problem has been studied previously with the assumptions that there is no strong damping term and the initial displacement and initial velocity have the same sign. However, from the physical point of view, it is obvious that the initial displacement and initial velocity may have different signs, and it is very necessary to consider the effects of the strong damping term. For example, the strong damping term indicates that the stress is not only proportional to the strain as with the Hooke law, but also proportional to the strain rate as in a linearized Kelvin–Voigt material. In this paper, by providing a completely different method from previous studies, we show that the solutions may blow up at infinity with arbitrary high initial energy when the model involves the strong damping term and the initial displacement and initial velocity may have different signs. Moreover, in this paper, we prove for the first time how to extend the solution over time (the whole half line) in studying the infinite time blow‐up phenomena for hyperbolic equations with logarithmic nonlinearity. These results fill in the gaps in previous studies on this type of models.

Semilinear pseudo-parabolic equations on manifolds with conical singularities

&lt;p style='text-indent:20px;'&gt;This paper studies the well-posedness of the semilinear pseudo-parabolic equations on manifolds with conical degeneration. By employing the Galerkin method and performing energy estimates, we first establish the local-in-time well-posedness of the solution. Moreover, to reveal the relationship between the initial datum and the global-in-time well-posedness of the solution we divide the initial datum into three classes by the potential well depth, i.e., the sub-critical initial energy level, the critical initial energy level and the sup-critical initial energy level (included in the arbitrary high initial energy case), and finally we give an affirmative answer to the question whether the solution exists globally or not. For the sub-critical and critical initial energy, thanks to the potential well theory, we not only obtain the invariant manifolds, global existence and asymptotic behavior of solutions, but also prove the finite time blow up of solutions and estimate the lower bound the of blowup time. For the sup-critical case, we show the assumptions for initial datum which cause the finite time blowup of the solution, realized by introducing a new auxiliary function. Additionally, we also provide some results concerning the estimates of the upper bound of the blowup time in the sup-critical initial energy.&lt;/p&gt;