On a parabolic equation in MEMS with fringing field

We consider the asymptotic behavior of the solutions to the equation ${u_{t}-u_{xx} = \lambda(1 + {\delta}u_{x}^{2})(1 - u)^{-2}}$ , which comes from Micro-Electromechanical Systems (MEMS) devices modeling. It is shown that when the fringing field exists (i.e., δ?> 0), there is a critical value λ _δ ^* > 0 such that if 0 < λ < λ _δ ^* , the equation has a global solution for some initial data; while for λ > λ _δ ^* , all solutions to the equation will quench at finite time. When the quenching happens, u has only finitely many quenching points for particular initial data. A one-side estimate is deduced for the quenching rate of u.     

Blow-up rates and uniqueness of large solutions for elliptic equations with nonlinear gradient term and singular or degenerate weights

This paper deals with the blow-up rate and uniqueness of large solutions of the elliptic equation ${\Delta u = b(x)f(u)+c(x)g(u)|\nabla u|^q}$ in ${\Omega \subset \mathbb{R}^N}$ , where q > 0, f(u) and g(u) are regularly varying functions at infinity, and the weight functions ${b(x),\,c(x) \in C^\alpha(\Omega,\,\mathbb{R}^+)}$ , 0 < α < 1, may be singular or degenerate on the boundary ${\partial\Omega}$ . Combining the regular variation theoretic approach of Cîrstea–R?dulescu and the systematic approach of Bandle–Giarrusso, we are able to improve and generalize most of the previously available results in the literature.     

Asymptotics for the minimization of a Ginzburg-Landau functional

LetΩ ? ?² be a smooth bounded simply connected domain. Consider the functional $$E_\varepsilon (u) = \frac{1}{2}\int\limits_\Omega {\left| {\nabla u} \right|^2 + \frac{1}{{4\varepsilon ^2 }}} \int\limits_\Omega {(|u|^2 - 1)^2 } $$ on the classH _g ¹ ={u εH ¹(Ω; ?);u=g on ?Ω} whereg:?Ω? → ? is a prescribed smooth map with ¦g¦=1 on ?Ω? and deg(g, ?Ω)=0. Let uu _ε be a minimizer for E_ε onH _g ¹ . We prove that u_ε → u₀ in \(C^{1,\alpha } (\bar \Omega )\) as ε → 0, where u₀ is identified. Moreover \(\left\| {u_\varepsilon - u_0 } \right\|_{L^\infty } \leqslant C\varepsilon ^2 \) .     

Chebyshev's inequalities for functions monotone in the mean

LetR be the reals ≥ 0. LetF be the set of mapsf: {1, 2, ?,n} →R. Choosew ∈ F withw _i = w(i) > 0. PutW _i = w₁ + ? + w_i. Givenf ∈ F, define \(\bar f\) ∈F by $$\bar f\left( i \right) = \frac{{\left\{ {w_i f\left( 1 \right) + \ldots + w_i f\left( i \right)} \right\}}}{{W_i }}.$$ Callf mean increasing if \(\bar f\) is increasing. Letf ₁, ?, f_t be mean decreasing andf _t+1,?: f_t+u be mean increasing. Put $$k = W_n^u \min \left\{ {w_i^{u - 1} W_i^{t - u} } \right\}.$$ Then $$k\mathop \sum \limits_{i = 1}^n w_i f_1 \left( i \right) \ldots f_{t + u} \left( i \right) \leqslant \mathop \prod \limits_{j = 1}^{t + u} (\mathop \sum \limits_{i = 1}^n w_i f_1 (i)).$$      

On the Laplace equation with a supercritical nonlinear Robin boundary condition in the half-space

We study the Laplace equation in the half-space ${\mathbb{R}_{+}^{n}}$ with a nonlinear supercritical Robin boundary condition ${\frac{\partial u}{\partial\eta }+\lambda u=u\left\vert u\right\vert^{\rho-1}+f(x)}$ on ${\partial \mathbb{R}_{+}^{n}=\mathbb{R}^{n-1}}$ , where n ≥ 3 and λ ≥ 0. Existence of solutions ${u \in E_{pq}= \mathcal{D}^{1, p}(\mathbb{R}_{+}^{n}) \cap L^{q}(\mathbb{R}_{+}^{n})}$ is obtained by means of a fixed point argument for a small data $f \in {L^{d}(\mathbb{R}^{n-1})}$ . The indexes p, q are chosen for the norm ${\Vert\cdot\Vert_{E_{pq}}}$ to be invariant by scaling of the boundary problem. The solution u is positive whether f(x) > 0 a.e. ${x\in\mathbb{R}^{n-1}}$ . When f is radially symmetric, u is invariant under rotations around the axis {x _n  = 0}. Moreover, in a certain L ^q -norm, we show that solutions depend continuously on the parameter λ ≥ 0.     

Stability of planar diffusion wave for nonlinear evolution equation

It is known that the one-dimensional nonlinear heat equation ut = f(u)x1x1,f’(u) > 0,u(±∞,t) = u±,u+ = u_ has a unique self-similar solution u(x1/1+t).In multi-dimensional space,u(x1/1+t) is called a planar diffusion wave.In the first part of the present paper,it is shown that under some smallness conditions,such a planar diffusion wave is nonlinearly stable for the nonlinear heat equation:ut-△f(u) = 0,x ∈ Rn.The optimal time decay rate is obtained.In the second part of this paper,it is further shown that this planar diffusion wave is still nonlinearly stable for the quasilinear wave equation with damping:utt + utt+ △f(u) = 0,x ∈ Rn.The time decay rate is also obtained.The proofs are given by an elementary energy method.     

Global existence for the Cauchy problem of the parabolic–parabolic Keller–Segel system on the plane

This paper is concerned with the Cauchy problem for the Keller–Segel system $$\left\{\begin{array}{l@{\quad}l}u_t = \nabla \cdot (\nabla u - u \nabla v) & \hbox{in } {\bf R}^{2} \times(0,\infty),\\v_t = \Delta v - \lambda v + u & \hbox{ in } {\bf R}^2 \times(0,\infty),\\u(x,0) = u_0 (x) \geq 0, \; v(x,0) = v_0 (x) \geq 0 & \hbox{ in} {\bf R}^2\end{array}\right.$$ with a constant λ ≥ 0, where ${(u_0, v_0) \in (L^1 ({\bf R}^2) \cap L^\infty ({\bf R}^2) ) \times (L^1 ({\bf R}^2) \cap H^1 ({\bf R}^2))}$ . Let $$m (u_0;{\bf R}^2) = \int\limits_{{\bf R}^2} u_0 (x) dx$$ . The same method as in [9] yields the existence of a blowup solution with m (u ₀; R ²) > 8π. On the other hand, it was recently shown in [7] that under additional hypotheses ${u_0 \log (1 + |x|^2) \in L^1 ({\bf R}^2)}$ and ${u_0 \log u_0 \in L^1 ({\bf R}^2)}$ , any solution with m(u ₀; R ²) < 8π exists globally in time. In[18], the extra assumptions were taken off, but the condition on mass was restricted to m (u ₀; R ²) < 4π. In this paper, we prove that any solution with m (u ₀; R ²) < 8π exists globally in time under no extra conditions. Furthermore the global existence of solutions is obtained under some condition on u ₀ also in the critical case m (u ₀; R ²) = 8π.     

Young measures generated by sequences in Morrey spaces

Let ${\Omega\subset\mathbb{R}^n}$ be open and bounded. For 1 ≤ p < ∞ and 0 ≤ λ < n, we give a characterization of Young measures generated by sequences of functions ${\{{\bf f}_j\}_{j=1}^\infty}$ uniformly bounded in the Morrey space ${L^{p,\lambda}(\Omega;\mathbb{R}^N)}$ with ${\{\left|{{\bf f}_j}\right|^p\}_{j=1}^\infty}$ equiintegrable. We then treat the case that each f _j = ? u _j for some ${{\bf u}_j\in W^{1,p}(\Omega;\mathbb{R}^N)}$ . As an application of our results, we consider the functional $${\bf u} \mapsto \int\limits_{\Omega}f({\bf x}, {\bf u}({\bf x}), {\bf {\nabla}}{\bf u}({\bf x})){\rm d}{\bf x},$$ and provide conditions that guarantee the existence of a minimizing sequence with gradients uniformly bounded in ${L^{p,\lambda}(\Omega;\mathbb{R}^{N\times n})}$ .     

On the stationary Navier–Stokes flows around a rotating body

Consider the stationary motion of an incompressible Navier–Stokes fluid around a rotating body $ \mathcal{K} = \mathbb{R}^3 \, \backslash \, {\Omega}$ which is also moving in the direction of the axis of rotation. We assume that the translational and angular velocities U, ω are constant and the external force is given by f = div F. Then the motion is described by a variant of the stationary Navier–Stokes equations on the exterior domain Ω for the unknown velocity u and pressure p, with U, ω, F being the data. We first prove the existence of at least one solution (u, p) satisfying ${\nabla u, p \in L_{3/2, \infty} (\Omega)}$ and ${u \in L_3, \infty (\Omega)}$ under the smallness condition on ${|U| + |\omega| + ||F||_{L_{3/2, \infty} (\Omega)}}$ . Then the uniqueness is shown for solutions (u, p) satisfying ${\nabla u, p \in L_{3/2, \infty} (\Omega) \cap L_{q, r} (\Omega)}$ and ${u \in L_{3, \infty} (\Omega) \cap L_{q*, r} (\Omega)}$ provided that 3/2 <? q <? 3 and ${{F \in L_{3/2, \infty} (\Omega) \cap L_{q, r} (\Omega)}}$ . Here L _q,r (Ω) denotes the well-known Lorentz space and q* =? 3q /(3 ? q) is the Sobolev exponent to q.     

Infinitely many small solutions for the p(x)-Laplacian operator with nonlinear boundary conditions

In this paper, we prove the existence of infinitely many small solutions to the following quasilinear elliptic equation ?Δ _p(x) u +  |u| ^p(x)-2 u =  f (x, u) in a smooth bounded domain Ω of ${\mathbb{R}^N}$ with nonlinear boundary conditions ${|\nabla u|^{p-2}\frac{\partial u}{\partial\nu} = |u|^{{q(x)-2}}u}$ . We also assume that ${\{q(x) = p^\ast(x)\}\neq \emptyset}$ , where p*(x) =  Np(x)/(N ? p(x)) is the critical Sobolev exponent for variable exponents. The proof is based on a new version of the symmetric mountain-pass lemma due to Kajikiya, and property of these solutions is also obtained.     

Regularity property of Donsker's delta function

LetL be the space of rapidly decreasing smooth functions on ? andL ^* its dual space. Let (L ²)⁺ and (L ²)^? be the spaces of test Brownian functionals and generalized Brownian functionals, respectively, on the white noise spaceL ^* with standard Gaussian measure. The Donsker delta functionδ(B(t)?x) is in (L ²)^? and admits the series representation $$\delta (B(t) - x) = (2\pi t)^{ - 1/2} \exp ( - x^2 /2t)\sum\limits_{n = 0}^\infty {(n!2^n )^{ - 1} H_n (x/\sqrt {2t} )} \times H_n (B(t)/\sqrt {2t} )$$ , whereH _n is the Hermite polynomial of degreen. It is shown that forφ in (L ²)⁺,g _t,φ(x)≡〈δ(B(t)?x), φ〉 is inL and the linear map takingφ intog _t,φ is continuous from (L ²)⁺ intoL. This implies that forf inL ^* is a generalized Brownian functional and admits the series representation $$f(B(t)) = (2\pi t)^{ - 1/2} \sum\limits_{n = 0}^\infty {(n!2^n )^{ - 1} \langle f,\xi _{n, t} \rangle } H_n (B(t)/\sqrt {2t} )$$ , whereξ _n,t is the Hermite function of degreen with parametert. This series representation is used to prove the Ito lemma forf inL ^*, $$f(B(t)) = f(B(u)) + \int_u^t {\partial _s^ * } f'(B(s)) ds + (1/2)\int_u^t {f''} (B(s)) ds$$ , where? _s ^* is the adjoint of \(\dot B(s)\) -differentiation operator? _s .     

The Cauchy problem for wave maps on a curved background

We consider the Cauchy problem for wave maps ${u: {\mathbb R}\times M \to N,}$ for Riemannian manifolds (M, g) and (N, h). We prove global existence and uniqueness for initial data, u[0]?=?(u ₀, u ₁), that is small in the critical norm ${\dot{H}^{\frac{d}{2}}\times \dot{H}^{\frac{d}{2}-1}(M; TN),}$ in the case (M, g) = ${({\mathbb R}^4, g),}$ where g is a small perturbation of the Euclidean metric. The proof follows the method introduced by Statah and Struwe in (Commun Pure Appl Math 47(5):719–754, 1994) for proving global existence and uniqueness of small data wave maps ${u : {\mathbb R}\times {\mathbb R}^d \to N}$ in the critical norm, for d?≥ 4. In our argument we employ the Strichartz estimates for variable coefficient wave equations established by Metcalfe and Tataru in (To appear in Math Ann).     

Long time small solutions to nonlinear parabolic equations

A sharp result on global small solutions to the Cauchy problem $$u_t = \Delta u + f\left( {u,Du,D^2 u,u_t } \right)\left( {t > 0} \right),u\left( 0 \right) = u_0 $$ In Rⁿ is obtained under the the assumption thatf is C^1+r forr>2/n and ‖u ₀‖C²(R ⁿ ) +‖u ₀‖W ₁ ² (R ⁿ ) is small. This implies that the assumption thatf is smooth and ‖u ₀ ‖W ₁ ^k (R ⁿ )+‖u ₀‖W ₂ ^k (R ⁿ ) is small fork large enough, made in earlier work, is unnecessary.     

Asymptotic expansions of solutions of equations with a deviating argument in Banach spaces

For the equation $$Lu = \frac{1}{i}\frac{{du}}{{dt}}\sum\nolimits_{j = 0}^m {A_j u} (l - h_j^0 - h_j^1 (t)) = f(t),$$ whereh ₀ ^o =0,h ₀ ¹ =0 (t) ≡ 0,h _j ^o = const > 0,h ₁ ^j (t),j= 1, ...,m are nonnegative continuously differentiable functions in [0, ∞), Aj are bounded linear operators, under conditions on the resolvent and on the right hand sidef(t), we have obtained an asymptotic formula for any solution u(t) from L₂ in terms of the exponential solutions u_k(t), k=1, ..., n, of the equation $$\frac{1}{i}\frac{{du}}{{dt}} - A_0 u - \sum\nolimits_{j = 0}^m {A_j u} (t - h_j^0 ) = 0,$$ connected with the poles λ_k, k=1, ..., n, of the resolvent R_λ in a certain strip.     

A class of integral equations and approximation of p-Laplace equations

Let ${\Omega\subset\mathbb{R}^n}$ be a bounded domain, and let 1 < p < ∞ and σ < p. We study the nonlinear singular integral equation $$ M[u](x) = f_0(x)\quad {\rm in}\,\Omega$$ with the boundary condition u = g ₀ on ?Ω, where ${f_0\in C(\overline\Omega)}$ and ${g_0\in C(\partial\Omega)}$ are given functions and M is the singular integral operator given by $$M[u](x)={\rm p.v.} \int\limits_{B(0,\rho(x))} \frac{p-\sigma}{|z|^{n+\sigma}}|u(x+z)-u(x)|^{p-2} (u(x+z)-u(x))\,{\rm dz},$$ with some choice of ${\rho\in C(\overline\Omega)}$ having the property, 0 < ρ(x) ≤ dist (x, ?Ω). We establish the solvability (well-posedness) of this Dirichlet problem and the convergence uniform on ${\overline\Omega}$ , as σ → p, of the solution u _σ of the Dirichlet problem to the solution u of the Dirichlet problem for the p-Laplace equation νΔ _p u = f ₀ in Ω with the Dirichlet condition u = g ₀ on ?Ω, where the factor ν is a positive constant (see (7.2)).     

On some new nonlocal solvability theorems for various classes of nonlinear differential equations

We study nonlinear boundary value problems of the form $$ [\Psi u']' + F(x;u',u) = g, u(0) = u(1) = 0 $$ , where Φ is a coercive continuous operator from L _p to L _q , and $$ F(x;u'',u',u) = g, u(0) = u(1) = 0 $$ ; first- and second-order partial differential equations $$ \Phi (x_1 ,x_2 ;u'_1 ,u'_2 ,u) = 0, \sum\limits_{i = 1}^\infty {[\Psi _i (u'_{x_i } )]'_{x_i } + F(x; \ldots ,u'_{x_i } , \ldots ,u) = g_i } $$ ; and general equations F(x; ..., u″ _ii , ...., ...., u′ _i , ...; u) = g(x) of elliptic type. We consider the corresponding boundary value problems of parabolic and hyperbolic type. The proof is based on various a priori estimates obtained in the paper and a nonlocal implicit function theorem.     

Planar nearrings on the Euclidean plane

Planar near-rings are generalized rings which can serve as coordinate domains for geometric structures in which each pair of nonparallel lines has a unique point of intersection. It is known that all planar nearrings can be constructed from regular groups of automorphisms of groups which can be viewed as the “action groups” of the planar nearring. In this article, we study planar nearrings whose additive group is \({(\mathbb{R}^n,+)}\) , in particular, n = 1 and 2. It is natural to study topological planar nearrings in this context, following ideas of the late Kenneth D. Magill, Jr. In the case of n = 1, we characterize all topological planar nearrings by their action groups \({(\mathbb{R}^*, \cdot)}\) or \({(\mathbb{R}^+, \cdot)}\) . For n = 2, these action groups and the circle group \({(\mathbb{U}, \cdot)}\) seem to be the most interesting cases, but the last case can be excluded completely. As a consequence, we obtain characterizations of the semi-homogeneous continuous mappings from \({\mathbb{R}^n}\) to \({\mathbb{R}}\) for n = 1 and 2. Such a mapping f enjoys the property that f(f(u)v) = f(u)f(v) for all \({u,v \in \mathbb{R}^n}\) . When \({f(\mathbb{R}^n) = \mathbb{R}^+}\) , f is a positive homogeneous mapping of degree 1.     

Remarks on eigenvalues and eigenfunctions of a special elliptic system

LetΛ ₁(Ω) be the first eigenvalue of the vector-valued problem $$\begin{gathered} \Delta u + \alpha grad div u + \Delta u = 0 in \Omega , \hfill \\ u = 0 in \partial \Omega , \hfill \\ \end{gathered} $$ , withα>0. Letλ ₁(Ω) be the first eigenvalue of the scalar problem $$\begin{gathered} \Delta u + \lambda u = 0 in \Omega , \hfill \\ u = 0 on \partial \Omega . \hfill \\ \end{gathered} $$ . The paper contains a proof of the inequality $$\left( {1 + \frac{\alpha }{n}} \right)\lambda _1 \left( \Omega \right) > \Lambda _1 \left( \Omega \right) > \left( \Omega \right)$$ and improves recent estimates of Sprössig [15] and Levine and Protter [11]. Moreover we show, ifΩ is a ball, that an eigensolution u₁, associated withΛ ₁(Ω) is not unique and that the eigensolutions for this and higher eigenvalues are never rotationally invariant. Finally we calculate some eigensolutions explicitly.     

A Matrix Differential Harnack Estimate for a Class of Ultraparabolic Equations

Let u be a positive solution of the ultraparabolic equation $$\partial _{t} u=\sum\limits_{i=1}^{n} \partial _{x_{i}}^{2} u+\sum\limits_{i=1}^{k} x_{i}\partial _{x_{n+i}}u \hspace {8mm} \text {on} \hspace {4mm} \mathbb {R}^{n+k}\times (0,T),$$ where 1 ≤ k ≤ n and 0 < T ≤ + ∞. Assume that u and its derivatives (w.r.t. the space variables) up to the second order are bounded on any compact subinterval of (0, T). Then the difference H(log u) ? H (log f) of the Hessian matrices of log u and of log f (both w.r.t. the space variables) is non-negatively definite, where f is the fundamental solution of the above equation with pole at the origin (0, 0). The estimate in the case n = k = 1 is due to Hamilton. As a corollary we get that \(\Delta l+\frac {n+3k}{2t}+\frac {6k}{t^{3}}\geq 0\) , where l = log u, and \(\Delta =\sum _{i=1}^{n+k} \partial _{x_{i}}^{2} \) .     

Semi-linear Wave Equations with Effective Damping

The authors study the Cauchy problem for the semi-linear damped wave equation $$u_{tt} - \Delta u + b\left( t \right)u_t = f\left( u \right), u\left( {0,x} \right) = u_0 \left( x \right), u_t \left( {0,x} \right) = u_1 \left( x \right)$$ in any space dimension n ≥ 1. It is assumed that the time-dependent damping term b(t) > 0 is effective, and in particular tb(t) → ∞ as t → ∞. The global existence of small energy data solutions for |f(u)| ≈ |u| ^p in the supercritical case of $p > \tfrac{2} {n}$ and $p \leqslant \tfrac{n} {{n - 2}}$ for n ≥ 3 is proved.