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In this paper we study the harmonic map heat flow on the euclidean space R d and we show an unconditional uniqueness result for maps with small initial data in the homogeneous Besov space B ˙ p , ∞ d p (R d) where d < p < ∞. As a consequence we obtain decay rates for solutions of the harmonic map flow of the form ‖ ∇ u (t) ‖ L ∞ (R d) ≤ C t − 1 2 . Additionally, under the assumption of a stronger spatial localization of the initial conditions, we show that the temporal decay happens in a self-similar way. We also explain that similar results hold for the biharmonic map heat flow and the semilinear heat equation with a power-type nonlinearity. [ABSTRACT FROM AUTHOR]

Let V be a bounded potential. We prove a Hölder stability estimate of determining the potential V from the boundary spectral data of the biharmonic operator Δ 2 + V. The boundary spectral data consists of the Dirichlet eigenvalues λ k and the normal derivatives of the eigenfunctions ∂ ν ϕ k , ∂ ν Δ ϕ k on the boundary of a bounded domain. The analysis depends on the asymptotic behavior of the spectral data. [ABSTRACT FROM AUTHOR]

In this paper, the following critical biharmonic elliptic problem { Δ 2 u = λ u + μ u ln ⁡ u 2 + | u | 2 ⁎ ⁎ − 2 u , x ∈ Ω , u = ∂ u ∂ ν = 0 , x ∈ ∂ Ω is considered, where Ω ⊂ R N is a bounded smooth domain with N ≥ 5. Some interesting phenomena occur due to the uncertainty on the sign of the logarithmic term. It is shown, mainly by using Mountain Pass Lemma, that the problem admits at least one nontrivial weak solution under some appropriate assumptions of λ and μ. Moreover, a nonexistence result is also obtained. Comparing the results in this paper with the known ones, one sees that some new phenomena occur when the logarithmic perturbation is introduced. [ABSTRACT FROM AUTHOR]

This paper is concerned with positive classical solutions u of the biharmonic equation Δ 2 u + u p = 0 in the punctured ball B 2 ﹨ { 0 } of R n , where n ≥ 5 and p > (n + 1) / (n − 3). We classify the isolated singularities of positive solutions and describe the asymptotic behavior of positive singular solutions without the sign assumption for − Δ u. [ABSTRACT FROM AUTHOR]

In this paper, we consider the well-posedness of the inhomogeneous nonlinear biharmonic Schrödinger equation with spatial inhomogeneity coefficient K (x) behaves like | x | − b for 0 < b < min ⁡ { N 2 , 4 }. We show the local well-posedness in the whole H s -subcritical case, with 0 < s ≤ 2. The difficulties of this problem come from the singularity of K (x) and the lack of differentiability of the nonlinear term. To resolve this, we derive the bilinear Strichartz's type estimates for the nonlinear biharmonic Schrödinger equations in Besov spaces. [ABSTRACT FROM AUTHOR]

We consider a two dimensional molecular beam epitaxy equation with biharmonic dissipation. We give an improved gradient bound, which removes the logarithmic dependence on the dissipation coefficient previously shown in [1]. [ABSTRACT FROM AUTHOR]

In this paper, we first prove some propositions of Sobolev spaces defined on a locally finite graph G = (V , E) , which are fundamental when dealing with equations on graphs under the variational framework. Then we consider a nonlinear biharmonic equation Δ 2 u − Δ u + (λ a + 1) u = | u | p − 2 u on G = (V , E). Under some suitable assumptions, we prove that for any λ > 1 and p > 2 , the equation admits a ground state solution u λ. Moreover, we prove that as λ → + ∞ , the solutions u λ converge to a solution of the equation { Δ 2 u − Δ u + u = | u | p − 2 u , in Ω , u = 0 , on ∂ Ω , where Ω = { x ∈ V : a (x) = 0 } is the potential well and ∂Ω denotes the boundary of Ω. [ABSTRACT FROM AUTHOR]

We obtain existence and multiplicity results for fourth order elliptic equations on R N involving u Δ (u 2) and sign-changing potentials. Our results generalize some recent results on this kind of problems. To study this kind of problems, we first consider the case that the potential V is coercive so that the working space can be compactly embedded into Lebesgue spaces. Then we studied the case that the potential V is bounded so that the working space is exactly H 2 (R N) , which can not be compactly embedded into Lebesgue spaces anymore. To deal with this more difficult case, we study the weak continuity of the term in the energy functional corresponding to the term u Δ (u 2) in the equation. [ABSTRACT FROM AUTHOR]

This paper is concerned with the degenerate initial–boundary value problem to the one-dimensional nonlinear wave equation u t t = ( ( 1 + u ) a u x ) x which arises in a number of various physical contexts. The global existence of smooth solutions to the degenerate problem was established under relaxed conditions on the initial–boundary data by the characteristic decomposition method. Moreover, we show that the solution is uniformly C 1 , α continuous up to the degenerate boundary and the degenerate curve is C 1 , α continuous for α ∈ ( 0 , min ⁡ { a 1 + a , 1 1 + a } ) . [ABSTRACT FROM AUTHOR]

In this paper, we find that the unbounded spaces Y σ ( R N ) ( 0 < σ < 2 m − 1 ) can provide the work spaces where complicated asymptotic behavior appears in the solutions of the Cauchy problem of the porous medium equation. To overcome the difficulties caused by the nonlinearity of the equation and the unbounded solutions, we establish the propagation estimates, the growth estimates and the weighted L 1 – L ∞ estimates for the solutions. [ABSTRACT FROM AUTHOR]

Positive singular radial entire solutions of a biharmonic equation with subcritical exponent are obtained via the entire radial solutions of the equation with supercritical exponent and the Kelvin's transformation. The expansions of such singular radial solutions at the singular point 0 are presented. Using these singular radial entire solutions, we construct solutions with a prescribed singular set for the Navier boundary value problem Δ 2 u = u p in Ω , u = Δ u = 0 on ∂ Ω where Ω is a smooth open set of R n with n ≥ 5 . [ABSTRACT FROM AUTHOR]

We consider a gradient flow modeling the epitaxial growth of thin films with slope selection. The surface height profile satisfies a nonlinear diffusion equation with biharmonic dissipation. We establish optimal local and global wellposedness for initial data with critical regularity. To understand the mechanism of slope selection and the dependence on the dissipation coefficient, we exhibit several lower and upper bounds for the gradient of the solution in physical dimensions d ≤ 3 . [ABSTRACT FROM AUTHOR]

We investigate a class of nonlinear biharmonic equations with p -Laplacian { Δ 2 u − β Δ p u + λ V ( x ) u = f ( x , u ) in R N , u ∈ H 2 ( R N ) , where N ≥ 1 , β ∈ R , λ > 0 is a parameter and Δ p u = div ( | ∇ u | p − 2 ∇ u ) with p ≥ 2 . Unlike most other papers on this problem, we replace Laplacian with p -Laplacian and allow β to be negative. Under some suitable assumptions on V ( x ) and f ( x , u ) , we obtain the existence and multiplicity of nontrivial solutions for λ large enough. The proof is based on variational methods as well as Gagliardo–Nirenberg inequality. [ABSTRACT FROM AUTHOR]

In this paper, by explicitly calculating the principal symbols of pseudodifferential operators, we establish two Weyl-type asymptotic formulas with sharp remainder estimates for the counting functions of the two classes of biharmonic Steklov eigenvalue problems of smooth bounded domains in a Riemannian manifold. [ABSTRACT FROM AUTHOR]

Using the dual method we establish the existence of a nodal ground state solution for the following class of problems { Δ 2 u = f ( u ) , in Ω , u = Bu = 0 , on ∂ Ω , where Ω ⊂ R N ( N ≥ 1 ) is a smooth bounded domain, Δ 2 is the biharmonic operator, B = Δ or B = ∂ ∂ ν and f is a C 1 -function having subcritical growth. [ABSTRACT FROM AUTHOR]

This work presents the construction of the existence theory of radial solutions to the elliptic equation Δ 2 u = ( − 1 ) k S k [ u ] + λ f ( x ) , x ∈ B 1 ( 0 ) ⊂ R N , provided either with Dirichlet boundary conditions u = ∂ n u = 0 , x ∈ ∂ B 1 ( 0 ) , or Navier boundary conditions u = Δ u = 0 , x ∈ ∂ B 1 ( 0 ) , where the k -Hessian S k [ u ] is the k -th elementary symmetric polynomial of eigenvalues of the Hessian matrix and the datum f ∈ L 1 ( B 1 ( 0 ) ) while λ ∈ R . We prove the existence of a Carathéodory solution to these boundary value problems that is unique in a certain neighborhood of the origin provided | λ | is small enough. Moreover, we prove that the solvability set of λ is finite, giving an explicity bound of the extreme value. [ABSTRACT FROM AUTHOR]

We derive the equation of a free vibrating thin plate whose mass is concentrated at the boundary, namely a Steklov problem for the biharmonic operator. We provide Hadamard-type formulas for the shape derivatives of the corresponding eigenvalues and prove that balls are critical domains under volume constraint. Finally, we prove an isoperimetric inequality for the first positive eigenvalue. [ABSTRACT FROM AUTHOR]

The aim of this work is to study the optimal exponent p to have solvability of problem { Δ 2 u = λ u | x | 4 + u p + c f in Ω , u > 0 in Ω , u = − Δ u = 0 on ∂ Ω , where p > 1 , λ > 0 , c > 0 , and Ω ⊂ R N , N > 4 , is a smooth and bounded domain such that 0 ∈ Ω . We consider f ⩾ 0 under some hypothesis that we will precise later. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we study radially symmetric entire solutions of We find the asymptotic behavior of these solutions for any and prove that for any there exists a solution with exactly linear growth. [Copyright &y& Elsevier]

Abstract: For a semilinear biharmonic equation with exponential nonlinearity, we study the existence and the asymptotic behavior of entire solutions. Furthermore, their stability and stability outside a compact set of () is discussed in any space dimension n. [Copyright &y& Elsevier]

Abstract: It is well known that the biharmonic equation with has positive solutions on if and only if the growth of the nonlinearity is critical or supercritical. We close a gap in the existing literature by proving the existence and uniqueness, up to scaling and symmetry, of oscillatory radial solutions on in the subcritical case. Analyzing the nodal properties of these solutions, we also obtain precise information about sign-changing large radial solutions and radial solutions of the Dirichlet problem on a ball. [Copyright &y& Elsevier]

Abstract: In this paper, under an improved Hardy–Rellich''s inequality, we study the existence of multiple and sign-changing solutions for a biharmonic equation in unbounded domain by the minimax method and linking theorem. [Copyright &y& Elsevier]

Abstract: In this paper, we study global positive solutions of the geometrically interesting equation: with in . We will establish several existence and non-existence theorems, including the classification result for with exactly linear growth condition. [Copyright &y& Elsevier]

Abstract: The aim of the article is to present a unified approach to the existence, uniqueness and regularity of solutions to problems belonging to a class of second order in time semilinear partial differential equations in Banach spaces. Our results are applied next to a number of examples appearing in literature, which fall into the class of strongly damped semilinear wave equations. The present work essentially extends the results on the existence and regularity of solutions to such problems. Previously, these problems have been considered mostly within the Hilbert space setting and with the main part operators being selfadjoint. In this article we present a more general approach, involving sectorial operators in reflexive Banach spaces. [Copyright &y& Elsevier]

Abstract: We investigate well posedness of the Cauchy problem for SG hyperbolic systems with non-smooth coefficients with respect to time. By assuming the coefficients to be Hölder continuous we show that this low regularity has a considerable influence on the behavior at infinity of the solution as well as on its regularity. This leads to well posedness in suitable Gelfand–Shilov classes of functions on . A simple example shows the sharpness of our results. [Copyright &y& Elsevier]

Abstract: The paper studies the longtime behavior of the Kirchhoff type equation with strong damping on . It shows that the related continuous semigroup possesses a global attractor which is connected and has finite fractal and Hausdorff dimension. [Copyright &y& Elsevier]

Abstract: This work investigates the existence of globally Lipschitz continuous solutions to a class of Cauchy problem of quasilinear wave equations. Applying Lax''s method and generalized Glimm''s method, we construct the approximate solutions of the corresponding perturbed Riemann problem and establish the global existence for the derivatives of solutions. Then, the existence of global Lipschitz continuous solutions can be carried out by showing the weak convergence of residuals for the source term of equation. [Copyright &y& Elsevier]

Abstract: We establish, subject to some natural additional assumptions imposed on the relation between the source and the damping, both well-posedness and effective optimal decay rates for the solutions of a semilinear model of the wave equation. The theory presented allows to consider both superlinear and sublinear behaviours of the dissipation in the presence of unstructured sources. [Copyright &y& Elsevier]

Abstract: We deal with the Cauchy problem for a strictly hyperbolic second-order operator with non-regular coefficients in the time variable. It is well-known that the problem is well-posed in in case of Lipschitz continuous coefficients and that the log-Lipschitz continuity is the natural threshold for the well-posedness in Sobolev spaces which, in this case, holds with a loss of derivatives. Here, we prove that any intermediate modulus of continuity between the Lipschitz and the log-Lipschitz one leads to an energy estimate with arbitrary small loss of derivatives. We also provide counterexamples to show that the following classification:is sharp [Copyright &y& Elsevier]

Abstract: The present paper concerns with the global structure and asymptotic behavior of the discontinuous solutions to flood wave equations. By solving a free boundary problem, we first obtain the global structure and large time behavior of the weak solutions containing two shock waves. For the Cauchy problem with a class of initial data, we use Glimm scheme to obtain a uniform BV estimate both with respect to time and the relaxation parameter. This yields the global existence of BV solution and convergence to the equilibrium equation as the relaxation parameter tends to 0. [Copyright &y& Elsevier]

We consider the Cauchy problem for the 2×2 strictly hyperbolic systemFor possibly large, discontinuous data, the continuous dependence of the solution with respect to both a0 and u0 is shown. Moreover, the solutions are characterized as unique limits of Kruǽkov''s entropic solutions constructed with regularized initial data aε0. [Copyright &y& Elsevier]