In this paper, we consider the second-order continuous time Galerkin approximation of the solution to the initial problem $u_t+{\int }_0^t\beta (t-s)Au\left(s\right)ds=0,u\left(0\right)=v,t>0,$ where A is an elliptic partial-differential operator and $\beta \left(t\right)$ is positive, nonincreasing and log-convex on $(0,\infty )$ with $0\le \beta \left(\infty \right)<\beta \left(0^{+}\right)\le \infty$. Error estimates are derived in the norm of $L_t^1(0,\infty ;L_x^2)$, and some estimates for the first order time derivatives of the errors are also given.

We present a novel approach for bounding the resolvent of $$H=-\Delta +i(A·\nabla +\nabla ·A)+V=:-\Delta +L1$$ for large energies. It is shown here that there exist a large integer $m$ and a large number ${\lambda }_0$ so that relative to the usual weighted $L^2$-norm, $$\parallel {\left(L{(-\Delta +(\lambda +i0))}^{-1}\right)}^m\parallel <\frac{1}{2}2$$ for all $\lambda >{\lambda }_0$. This requires suitable decay and smoothness conditions on $A,V$. The estimate (2) is trivial when $A=0$, but difficult for large $A$ since the gradient term exactly cancels the natural decay of the free resolvent. To obtain (2), we introduce a conical decomposition of the resolvent and...

We consider “nonconventional” averaging setup in the form $\frac{\mathrm{d}{X}^{\epsilon }\left(t\right)}{\mathrm{d}t}=\epsilon B(X^{\epsilon }\left(t\right)$, $𝛯\left(q_1\left(t\right)\right),𝛯\left(q_2\left(t\right)\right),...,𝛯\left(q_{\ell }\left(t\right)\right))$ where $𝛯\left(t\right)$, $t\ge 0$ is either a stochastic process or a dynamical system with sufficiently fast mixing while $q_j\left(t\right)={\alpha }_{j}t$, ${\alpha }_{1}lt;{\alpha }_{2}lt;\cdots lt;{\alpha }_k$ and $q_j$, $j=k+1,...,\ell$ grow faster than linearly. We show that the properly normalized error term in the “nonconventional” averaging principle is asymptotically Gaussian.

It is known that the nonlinear nonhomogeneous backward Cauchy problem $u_t\left(t\right)+Au\left(t\right)=f(t,u\left(t\right))$, $0\le t<\tau$ with $u\left(\tau \right)=\phi$, where $A$ is a densely defined positive self-adjoint unbounded operator on a Hilbert space, is ill-posed in the sense that small perturbations in the final value can lead to large deviations in the solution. We show, under suitable conditions on $\phi$ and $f$, that a solution of the above problem satisfies an integral equation involving the spectral representation of $A$, which is also ill-posed. Spectral truncation...

We consider the quasilinear equation ${\Delta }_{p}u+K\left(\right|x\left|\right)u^q=0$, and present the proof of the local existence of positive radial solutions near $0$ under suitable conditions on $K$. Moreover, we provide a priori estimates of positive radial solutions near $\infty$ when $r^{-\ell }K\left(r\right)$ for $\ell \ge -p$ is bounded near $\infty$.

It is shown that there is a subspace $Z_q$ of ${\ell }_q$ for $1<q<2$ which is isomorphic to ${\ell }_q$ such that ${\ell }_q/Z_q$ does not have the approximation property. On the other hand, for $2<p<\infty$ there is a subspace $Y_p$ of ${\ell }_p$ such that $Y_p$ does not have the approximation property (AP) but the quotient space ${\ell }_p/Y_p$ is isomorphic to ${\ell }_p$. The result is obtained by defining random “Enflo-Davie spaces” $Y_p$ which with full probability fail AP for all $2<p\le \infty$ and have AP for all $1\le p\le 2$. For $1<p\le 2$, $Y_p$ are isomorphic to ${\ell }_p$.

Let $\Omega$ be a bounded domain of class $C^2$ in $ℝ$N and let $K$ be a compact subset of $\partial \Omega$. Assume that $q\ge (N+1)/\left(N-1\right)$ and denote by $U_K$ the maximal solution of $-\Delta u+u^q=0$ in $\Omega$ which vanishes on $\partial \Omega \setminus K$. We obtain sharp upper and lower estimates for $U_K$ in terms of the Bessel capacity $C_{2/q,q^{\text{'}}}$ and prove that $U_K$ is $\sigma$-moderate. In addition we describe the precise asymptotic behavior of $U_K$ at points $\sigma \in K$, which depends on the “density” of $K$ at $\sigma$, measured in terms of the capacity $C_{2/q,q^{\text{'}}}$.

We study the superconvergence of the finite volume method for a nonlinear elliptic problem using linear trial functions. Under the condition of $C$-uniform meshes, we first establish a superclose weak estimate for the bilinear form of the finite volume method. Then, we prove that on the mesh point set $S$, the gradient approximation possesses the superconvergence: ${max}_{P\in S}|(\nabla u-\overline{\nabla }{u}_h)\left(P\right)|=O\left(h^2\right){|lnh|}^{3/2}$, where $\overline{\nabla }$ denotes the average gradient on elements containing vertex $P$. Furthermore, by using the interpolation post-processing...

We consider solutions of quasilinear equations $u_t=\Delta u^m+u^p$ in ${ℝ}^N$ with the initial data $u_0$ satisfying $0<u_0<M$ and ${lim}_{\left|x\right|\to \infty }{u}_0\left(x\right)=M$ for some constant $M>0$. It is known that if $0<m<p$ with $p>1$, the blow-up set is empty. We find solutions $u$ that blow up throughout ${ℝ}^N$ when $m>p>1$.

We investigate the average behavior of the $n$th normalized Fourier coefficients of the $j$th ($j\ge 2$ be any fixed integer) symmetric power $L$-function (i.e., $L(s,{\mathrm{sym}}^{j}f)$), attached to a primitive holomorphic cusp form $f$ of weight $k$ for the full modular group $SL(2,\mathbb{Z})$ over certain sequences of positive integers. Precisely, we prove an asymptotic formula with an error term for the sum $$S_j^{*}:=\sum _{a_1^2+a_2^2+a_3^2+a_4^2+a_5^2+a_6^2\le x(a_1,a_2,a_3,a_4,a_5,a_6)\in {\mathbb{Z}}^6}{\lambda }_{{\mathrm{sym}}^{j}f}^2(a_1^2+a_2^2+a_3^2+a_4^2+a_5^2+a_6^2),$$ where $x$ is sufficiently large, and $$L(s,{\mathrm{sym}}^{j}f):=\sum _{n=1}^{\infty }\frac{{\lambda }_{{\mathrm{sym}}^{j}f}\left(n\right)}{n^s}.$$ When $j=2$, the error term which we obtain improves the earlier known result.

Let $\alpha \&gt;1$ be irrational. Several authors studied the numbers $${\ell }^m\left(\alpha \right)=inf\left\{\right|y|:y\in {\Lambda }_m,y\ne 0\},$$ where $m$ is a positive integer and ${\Lambda }_m$ denotes the set of all real numbers of the form $y={ϵ}_0{\alpha }^n+{ϵ}_1{\alpha }^{n-1}+\cdots +{ϵ}_{n-1}\alpha +{ϵ}_n$ with restricted integer coefficients $|{ϵ}_i|\le m$. The value of ${\ell }^1\left(\alpha \right)$ was determined for many particular Pisot numbers and ${\ell }^m\left(\alpha \right)$ for the golden number. In this paper the value of ${\ell }^m\left(\alpha \right)$ is determined for irrational numbers $\alpha$, satisfying ${\alpha }^2=a\alpha \pm 1$ with a positive integer $a$.

We consider $2m$th-order $(m\ge 2)$ semilinear parabolic equations $u_t=-{\left(-\Delta \right)}^{m}u\pm {\left|u\right|}^{p-1}u$ in ${ℛ}^N\times {ℝ}_{+}$ $(p>1)$, with Dirac’s mass $\delta \left(x\right)$ as the initial function. We show that for $p<p_0=1+2m/N$, the Cauchy problem admits a solution $u(x,t)$ which is bounded and smooth for small $t>0$, while for $p\ge p_0$ such a local in time solution does not exist. This leads to a boundary layer phenomenon in constructing a proper solution via regular approximations.

Consider the family uₜ = Δu + G(u), t > 0, $x\in {\Omega }_{\epsilon }$, ${\partial }_{{\nu }_{\epsilon }}u=0$, t > 0, $x\in \partial {\Omega }_{\epsilon }$, $\left(E_{\epsilon }\right)$ of semilinear Neumann boundary value problems, where, for ε > 0 small, the set ${\Omega }_{\epsilon }$ is a thin domain in ${ℝ}^l$, possibly with holes, which collapses, as ε → 0⁺, onto a (curved) k-dimensional submanifold of ${ℝ}^l$. If G is dissipative, then equation $\left(E_{\epsilon }\right)$ has a global attractor ${}_{\epsilon }$. We identify a “limit” equation for the family $\left(E_{\epsilon }\right)$, prove convergence of trajectories and establish an upper semicontinuity result for the family ${}_{\epsilon }$ as ε → 0⁺. ...

This paper considers the initial-boundary value problem for the nonlinear diffusion equation with the perturbation term $$u_t+(-\Delta +1)\beta \left(u\right)+G\left(u\right)=g\text{in}\Omega \times (0,T)$$ in an unbounded domain $\Omega \subset {ℝ}^N$ with smooth bounded boundary, where $N\in \mathbb{N}$, $T>0$, $\beta$, is a single-valued maximal monotone function on $ℝ$, e.g., $$\beta \left(r\right)={\left|r\right|}^{q-1}r(q>0,q\ne 1)$$ and $G$ is a function on $ℝ$ which can be regarded as a Lipschitz continuous operator from ${\left(H^1\left(\Omega \right)\right)}^{*}$ to ${\left(H^1\left(\Omega \right)\right)}^{*}$. The present work establishes existence and estimates for the above problem.