Regularity of the extremal solution for some elliptic problems with singular nonlinearity and advection

In this note, we investigate the regularity of the extremal solution u^? for the semilinear elliptic equation −△u+c(x)⋅∇u=λf(u) on a bounded smooth domain of Rn with Dirichlet boundary condition. Here f is a positive nondecreasing convex function, exploding at a finite value a∈(0,∞). We show that the extremal solution is regular in the low-dimensional case. In particular, we prove that for the radial case, all extremal solutions are regular in dimension two.     

On decay and blow-up of the solution for a viscoelastic wave equation with boundary damping and source terms

In this paper we consider the decay and blow-up properties of a viscoelastic wave equation with boundary damping and source terms. We first extend the decay result (for the case of linear damping) obtained by Lu et al. (On a viscoelastic equation with nonlinear boundary damping and source terms: Global existence and decay of the solution, Nonlinear Analysis: Real World Applications 12 (1) (2011), 295-303) to the nonlinear damping case under weaker assumption on the relaxation function g(t). Then, we give an exponential decay result without the relation between g^′(t) and g(t) for the linear damping case, provided that ‖g‖_L¹(0,∞) is small enough. Finally, we establish two blow-up results: one is for certain solutions with nonpositive initial energy as well as positive initial energy for both the linear and nonlinear damping cases, the other is for certain solutions with arbitrarily positive initial energy for the linear damping case.     

A non-local regularization of first order Hamilton-Jacobi equations

In this paper, we investigate the regularizing effect of a non-local operator on first-order Hamilton-Jacobi equations. We prove that there exists a unique solution that is C² in space and C¹ in time. In order to do so, we combine viscosity solution techniques and Green's function techniques. Viscosity solution theory provides the existence of a W^1,∞ solution as well as uniqueness and stability results. A Duhamel's integral representation of the equation involving the Green's function permits to prove further regularity. We also state the existence of C^∞ solutions (in space and time) under suitable assumptions on the Hamiltonian. We finally give an error estimate in L^∞ norm between the viscosity solution of the pure Hamilton-Jacobi equation and the solution of the integro-differential equation with a vanishing non-local part.     

Global conservative solutions of the generalized hyperelastic-rod wave equation

We prove existence of global and conservative solutions of the Cauchy problem for the nonlinear partial differential equation where f is strictly convex or concave and g is locally uniformly Lipschitz. This includes the Camassa-Holm equation (f(u)=u²/2 and g(u)=κu+u²) as well as the hyperelastic-rod wave equation (f(u)=γu²/2 and g(u)=(3−γ)u²/2) as special cases. It is shown that the problem is well-posed for initial data in H¹(R) if one includes a Radon measure that corresponds to the energy of the system with the initial data. The solution is energy preserving. Stability is proved both with respect to initial data and the functions f and g. The proof uses an equivalent reformulation of the equation in terms of Lagrangian coordinates.     

Regularity, monotonicity and symmetry of positive solutions of m-Laplace equations

We consider the Dirichlet problem for positive solutions of the equation −Δ_m(u)=f(u) in a bounded smooth domain Ω, with f locally Lipschitz continuous, and prove some regularity results for weak solutions. In particular when f(s)>0 for s>0 we prove summability properties of , and Sobolev's and Poincaré type inequalities in weighted Sobolev spaces with weight |Du|^m−2. The point of view of considering |Du|^m−2 as a weight is particularly useful when studying qualitative properties of a fixed solution. In particular, exploiting these new regularity results we can prove a weak comparison principle for the solutions and, using the well known Alexandrov-Serrin moving plane method, we then prove a general monotonicity (and symmetry) theorem for positive solutions u of the Dirichlet problem in bounded (and symmetric in one direction) domains when f(s)>0 for s>0 and m>2. Previously, results of this type in general bounded (and symmetric) domains had been proved only in the case 1<m<2.     

Large time behavior for a viscous Hamilton-Jacobi equation with Neumann boundary condition

We prove the existence and the uniqueness of strong solutions for the viscous Hamilton-Jacobi equation: with Neumann boundary condition, and initial data μ₀, a continuous function. The domain Ω is a bounded and convex open set with smooth boundary, a∈R,a≠0 and p>0. Then, we study the large time behavior of the solution and we show that for p∈(0,1), the extinction in finite time of the gradient of the solution occurs, while for p?1 the solution converges uniformly to a constant, as t→∞.     

Blowup stability of solutions of the nonlinear heat equation with a large life span

We study the Cauchy problem for the nonlinear heat equation u_t-?u=|u|^p-1u in R^N. The initial data is of the form u₀=λ?, where ?∈C₀(R^N) is fixed and λ>0. We first take 1<p<p_f, where p_f is the Fujita critical exponent, and ?∈C₀(R^N)∩L¹(R^N) with nonzero mean. We show that u(t) blows up for λ small, extending the H. Fujita blowup result for sign-changing solutions. Next, we consider 1<p<p_s, where p_s is the Sobolev critical exponent, and ?(x) decaying as |x|^-σ at infinity, where p<1+2/σ. We also prove that u(t) blows up when λ is small, extending a result of T. Lee and W. Ni. For both cases, the solution enjoys some stable blowup properties. For example, there is single point blowup even if ? is not radial.     

Long-time behavior of reaction-diffusion equations with dynamical boundary condition

In this paper, we study the long-time behavior of the reaction-diffusion equation with dynamical boundary condition, where the nonlinear terms f and g satisfy the polynomial growth condition of arbitrary order. Some asymptotic regularity of the solution has been proved. As an application of the asymptotic regularity results, we can not only obtain the existence of a global attractor A in (H¹(Ω)∩L^p(Ω))×L^q(Γ) immediately, but also can show further that A attracts every L²(Ω)×L²(Γ)-bounded subset with (H¹(Ω)∩L^p+δ(Ω))×L^q+κ(Γ)-norm for any δ,κ∈[0,∞).     

Homoclinic bifurcations at the onset of pulse self-replication

We establish a series of properties of symmetric, N-pulse, homoclinic solutions of the reduced Gray-Scott system: u^″=uv², v^″=v−uv², which play a pivotal role in questions concerning the existence and self-replication of pulse solutions of the full Gray-Scott model. Specifically, we establish the existence, and study properties, of solution branches in the (α,β)-plane that represent multi-pulse homoclinic orbits, where α and β are the central values of u(x) and v(x), respectively. We prove bounds for these solution branches, study their behavior as α→∞, and establish a series of geometric properties of these branches which are valid throughout the (α,β)-plane. We also establish qualitative properties of multi-pulse solutions and study how they bifurcate, i.e., how they change along the solution branches.     

Nodal solutions for singularly perturbed equations with critical exponential growth

The existence and concentration behavior of nodal solutions are established for the equation −?²Δu+V(z)u=f(u) in Ω, where Ω is a domain in R², not necessarily bounded, V is a positive Hölder continuous function and f∈C¹ is an odd function having critical exponential growth.     

Weighted pseudo almost periodic mild solutions of semilinear evolution equations with nonlocal conditions

In this paper, applying the theory of semigroups of operators to evolution families and Banach fixed point theorem, we prove the existence and uniqueness of the weighted pseudo almost periodic mild solution of the semilinear evolution equation x^′(t)=A(t)x(t)+f(t,x(t)) with nonlocal conditions x(0)=x₀+g(x) in Banach space X under some suitable hypotheses.     

Global and periodic solutions for nonlinear wave equations with some localized nonlinear dissipation

We discuss the existence of global or periodic solutions to the nonlinear wave equation with the boundary condition , where Ω is a bounded domain in R^N,ρ(x,v) is a function like ρ(x,v)=a(x)g(v) with g′(v)?0 and β(x,u) is a source term of power nonlinearity. a(x) is assumed to be positive only in a neighborhood of a part of the boundary ∂Ω and the stability property is very delicate, which makes the problem interesting.     

Existence of solutions for degenerate quasilinear elliptic equations

This paper deals with a class of degenerate quasilinear elliptic equations of the form −div(a(x,u,∇u)=g−div(f), where a(x,u,∇u) is allowed to be degenerate with the unknown u. We prove existence of bounded solutions under some hypothesis on f and g. Moreover we prove that there exists a renormalized solution in the case where g∈L¹(Ω) and f∈(L^p′(Ω))^N.     

The heterogeneous Allen-Cahn equation in a ball: Solutions with layers and spikes

Let u? be a single layered radially symmetric unstable solution of the Allen-Cahn equation −?²Δu=u(u−a(|x|))(1−u) over the unit ball with Neumann boundary conditions. Based on our estimate of the small eigenvalues of the linearized eigenvalue problem at u? when ? is small, we construct solutions of the form u?+v?, with v? non-radially symmetric and close to zero in the unit ball except near one point x₀ such that |x₀| is close to a nondegenerate critical point of a(r). Such a solution has a sharp layer as well as a spike.     

Parabolic equations of Von Karman type on Kähler manifolds, II

We complete the study of a parabolic version of a system of Von Karman type on a compact Kähler manifold of complex dimension m. We consider a family of problems k(P). We prove existence of local in time solutions when k=0. When −m?k<0, we define a notion of weak solution, and give some uniqueness and existence results.     

A note on parabolic equation with nonlinear dynamical boundary condition

We consider a semilinear parabolic equation subject to a nonlinear dynamical boundary condition that is related to the so-called Wentzell boundary condition. First we prove the existence and uniqueness of global solutions as well as the existence of a global attractor. Then we derive a suitable ?ojasiewicz-Simon type inequality to show the convergence of global solutions to single steady states as time tends to infinity under the assumption that the nonlinear terms f,g are real analytic. Moreover, we provide an estimate for the convergence rate.     

A note on scalar curvature type problems in dimension 4

Let (M,g) be a 4-dimensional compact Riemannian manifold and let a,f be positive smooth functions on M. In this note, we prove that the problem Δgu+a(x)u=f(x)u³ always admits a positive solution, up to a conformal deformation of g. This leads to a geometric obstruction result for the prescribed scalar curvature problem.     

The Gradient Flow of <Emphasis Type="Italic">∫</Emphasis><Subscript><Emphasis Type="Italic">M</Emphasis></Subscript>|Rm|<Superscript>2</Superscript>

We study the gradient flow of the Riemannian functional ℱ(g):=∫ _M |Rm|². This flow corresponds to a fourth-order degenerate parabolic equation for a Riemannian metric. We prove that the degeneracies may be accounted for entirely by diffeomorphism flow, and hence we show short-time existence using the DeTurck method. We prove L ² derivative estimates of Bernstein-Bando-Shi type and use these to give a basic obstruction to long time existence and prove a compactness theorem.      

Critical exponents and blow-up rate for a nonlinear diffusion equation with logarithmic boundary flux

In this paper, we establish the critical global existence exponent and the critical Fujita exponent for the nonlinear diffusion equation u_t=(log^σ(1+u)u_x)_x, in R₊×(0,+∞), subject to a logarithmic boundary flux , furthermore give the blow-up rate for the nonglobal solutions.     

Dirichlet inhomogeneous boundary value problem for the n+1 complex Ginzburg-Landau equation

We study the following complex Ginzburg-Landau equation with cubic nonlinearity on for under initial and Dirichlet boundary conditions u(x,0)=h(x) for x∈Ω, u(x,t)=Q(x,t) on ∂Ω where h,Q are given smooth functions. Under suitable conditions, we prove the existence of a global solution in H¹. Further, this solution approaches to the solution of the NLS limit under identical initial and boundary data as a,b→0⁺.