Considered here is the first initial boundary value problem for a semilinear degenerate parabolic equation involving the Grushin operator in a bounded domain $\Omega$. We prove the regularity and exponential growth of a pullback attractor in the space $S_0^2\left(\Omega \right)\cap L^{2p-2}\left(\Omega \right)$ for the nonautonomous dynamical system associated to the problem. The obtained results seem to be optimal and, in particular, improve and extend some recent results on pullback attractors for reaction-diffusion equations in bounded domains.

The Kato theorem for abstract differential equations is applied to establish the local well-posedness of the strong solution for a nonlinear generalized Novikov equation in space $C\left(\right[0,T),H^s(R\left)\right)\cap C^1\left(\right[0,T),H^{s-1}(R\left)\right)$ with $s>(3/2)$. The existence of weak solutions for the equation in lower-order Sobolev space $H^s\left(R\right)$ with $1\le s\le (3/2)$ is acquired.

By establishing a maximal principle and constructing upper and lower solutions, the existence of positive solutions for the eigenvalue problem of a class of fractional differential equations is discussed. Some sufficient conditions for the existence of positive solutions are established.

A class of stage-structured predator-prey model with time delay and delay-dependent parameters is considered. Its linear stability is investigated and Hopf bifurcation is demonstrated. Using normal form theory and center manifold theory, some explicit formulae for determining the stability and the direction of the Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations are obtained. Finally, numerical simulations are performed to verify the analytical results.

Numerical solutions for Burgers’ equation based on the Galerkins’ method using cubic B-splines as both weight and interpolation functions are set up. It is shown that this method is capable of solving Burgers’ equation accurately for values of viscosity ranging from very small to large. Three standard problems are used to validate the proposed algorithm. A linear stability analysis shows that a numerical scheme based on a Cranck-Nicolson approximation in time is unconditionally stable.

We investigate the existence and uniqueness of solutions to the nonlocal boundary value problem for nonlinear impulsive fractional differential equations of order $\alpha \in (2,3]$. By using some well-known fixed point theorems, sufficient conditions for the existence of solutions are established. Some examples are presented to illustrate the main results.

We study the singular fractional-order boundary-value problem with a sign-changing nonlinear term , ${𝒟}^{{\mu }_{n-1}}x(1)={\sum }_{j=1}^{p-2}{a}_j{𝒟}^{{\mu }_{n-1}}x({\xi }_j)$, where , $n\in \mathbb{N}$ and $n\ge 3$ with and , satisfying , ${𝒟}^{\alpha }$ is the standard Riemann-Liouville derivative, $f:[\mathrm{0,1}]\times {ℝ}^n\to ℝ$ is a sign-changing continuous function and may be unbounded from below with respect to $x_i$, and $p:(\mathrm{0,1})\to [0,\infty )$ is continuous. Some new results on the existence of nontrivial solutions for the above problem are obtained by computing the topological degree of a completely continuous field.

The existence of at least one positive solution is established for a class of semipositone fractional differential equations with Riemann-Stieltjes integral boundary condition. The technical approach is mainly based on the fixed-point theory in a cone.

We will present a generalization of Mahadevan’s version of theKrein-Rutman theorem for a compact, positively 1-homogeneous operator on aBanach space having the properties of being increasing with respect to a cone $P$ and such that there is a nonzero $u\in P\setminus \{\theta \}-P$ for which $M{T}^{p}u\ge u$ for somepositive constant $M$ and some positive integer p. Moreover, we give some newresults on the uniqueness of positive eigenvalue with positive eigenfunction andcomputation of the fixed point index. As applications, the existence of positivesolutions for p-Laplacian boundary-value problems is considered under someconditions concerning the positive eigenvalues corresponding to the relevantpositively 1-homogeneous operators.

The pseudoparabolic regularization technique is employed to study the local well-posedness of strong solutions for a nonlinear dispersive model, which includes the famous Camassa-Holm equation. The local well-posedness is established in the Sobolev space $H^s(R)$ with $s>3/2$ via a limiting procedure.

We investigate the existence and uniqueness of positive solutions of the following nonlinear fractional differential equation with integral boundary value conditions${}^C{D}^{\alpha }u(t)+f(t,u(t))=0$, , $u(0)=u^{\mathrm{\prime \prime }}(0)=0,u(1)=\lambda {\int }_0^{1}u(s)ds$, where , and ${}^C{D}^{\alpha }$ is the Caputo fractional derivative and $f:[\mathrm{0,1}]\times [0,\infty )\to [0,\infty )$ is a continuous function. Our analysis relies on a fixed point theorem in partially ordered sets. Moreover, we compare our results with others that appear in the literature.

A modification of homotopy analysis method (HAM) known as spectral homotopy analysis method (SHAM) is proposed to solve linear Volterra integrodifferential equations. Some examples are given in order to test the efficiency and the accuracy of the proposed method. The SHAM results show that the proposed approach is quite reasonable when compared to SHAM results and exact solutions.

We study the multiplicity of solutions for the following fractional boundary value problem: $\mathrm{(d}/d\mathrm{t)}((1/\mathrm{2)}{\hspace{0.17em}}_0{D}_t^{-\beta }(u\text{'}(t))+\mathrm{(1}/\mathrm{2)}{\hspace{0.17em}}_0{D}_T^{-\beta }(u\text{'}(t)))+\lambda \nabla F(t,u(t))=0,\text{a.e.}t\in [0,T],u(0)=u(T)=0,$ where ${\hspace{0.17em}}_0{D}_t^{-\beta }$ and ${\hspace{0.17em}}_0{D}_T^{-\beta }$ are the left and right Riemann-Liouville fractional integrals of order , respectively, $\lambda >0$ is a real number, $F:[0,T]\times {ℝ}^N\to ℝ$ is a given function, and $\nabla F(t,x)$ is the gradient of $F$ at $x$. The approach used in this paper is the variational method. More precisely, the Weierstrass theorem and mountain pass theorem are used to prove the existence of at least two nontrivial solutions.

We study nonlinear impulsive differential equations of fractional order with irregular boundary conditions. Some existence and uniqueness results are obtained by applying standard fixed-point theorems. For illustration of the results, some examples are discussed.

By establishing the corresponding variational framework and using the mountain pass theorem, linking theorem, and Clark theorem in critical point theory, we give the existence of multiple solutions for a fractional difference boundary value problem with parameter. Under some suitable assumptions, we obtain some results which ensure the existence of a well precise interval of parameter for which the problem admits multiple solutions. Some examples are presented to illustrate the main results.

We investigate the blow-up properties of the positive solution of the Cauchy problem for a quasilinear degenerate parabolic equation with strongly nonlinear source $u_t=div(|\nabla u^m{|}^{p-2}\nabla u^l)+u^q,(x$ $,t)\in R^N\times (0,T)$, where $N\ge 1$, $p>2$ , and $m$, $l$,$q>1$, and give a secondary critical exponent on the decay asymptotic behavior of an initial value at infinity for the existence and nonexistence of global solutions of the Cauchy problem. Moreover, under some suitable conditions we prove single-point blow-up for a large class of radial decreasing solutions.

We study the following nonhomogeneous $A$-harmonic equations: $d^{*}A(x,du(x))+B(x,u(x))=0,x\in \mathrm{\Omega },u(x)=0,x\in \partial \mathrm{\Omega }$, where $\mathrm{\Omega }\subset {ℝ}^n$ is a bounded and convex Lipschitz domain, $A(x,du(x))$ and $B(x,u(x))$ satisfy some $p(x)$-growth conditions, respectively. We obtain the existence of weak solutions for the above equations in subspace ${𝔎}_0^{1,p(x)}(\mathrm{\Omega },{\mathrm{\Lambda }}^{l-\mathrm{1}})$ of $W_0^{1,p(x)}(\mathrm{\Omega },{\mathrm{\Lambda }}^{l-\mathrm{1}})$.

Splitting methods have recently received much attention due to the fact that many nonlinear problems arising in applied areas such as image recovery, signal processing, and machine learning are mathematically modeled as a nonlinear operator equation and this operator is decomposed as the sum of two (possibly simpler) nonlinear operators. Most of the investigation on splitting methods is however carried out in the framework of Hilbert spaces. In this paper, we consider these methods in the setting of Banach spaces. We shall introduce two iterative forward-backward splitting methods with relaxations and errors to find zeros of the sum of two accretive operators in the Banach spaces. We shall prove the weak and strong convergence of these methods under mild conditions. We also discuss applications of these methods to variational inequalities, the split feasibility problem, and a constrained convex minimization problem.

We consider the existence of single and multiple positive solutions for a second-order Sturm-Liouville boundary value problem in a Banach space. The sufficient condition for the existence of positive solution is obtained by the fixed point theorem of strict set contraction operators in the frame of the ODE technique. Our results significantly extend and improve many known results including singular and nonsingular cases.

By means of the fixed point theory in cones, we investigate the existence of positive solutions for the following second-order singular differential equations with a negatively perturbed term: $-u^{\prime \prime }\left(t\right)=\lambda \left[f\left(t,u\left(t\right)\right)-q\left(t\right)\right]$, , $\alpha u\left(0\right)-\beta u^{\prime }\left(0\right)={\displaystyle {\int }_0^{1}u\left(s\right)d\xi \left(s\right)}$, $\gamma u\left(1\right)+\delta u^{\prime }\left(1\right)={\displaystyle {\int }_0^{1}u\left(s\right)d\eta (s),}$where $\lambda >0$ is a parameter; $f:\left(0,1\right)\times \left(0,\infty \right)\to \left[0,\infty \right)$ is continuous; $f\left(t,x\right)$ may be singular at $t=0$, $t=\mathrm{1,}$ and $x=0$, and the perturbed term $q:\left(0,1\right)\to \left[0,+\infty \right)$ is Lebesgue integrable and may have finitely many singularities in $\left(0,1\right)$, which implies that the nonlinear term may change sign.

The $L^2$ space solution of an initial boundary problem for a generalizeddamped Boussinesq equation is constructed. Certain assumptions on the coefficients ofthe equation are found to show the existence and uniqueness of the global solution to theinitial boundary problem. The explicit expression for the large time asymptotic solution isobtained.

We study the degenerate semilinear elliptic systems of the form $-\text{div}\left(h_1\right(x)\nabla u)=$ $\lambda \left(a\right(x)u+b(x\left)v\right)+F_u(x,u,v),x\in \Omega ,-\text{div}\left(h_2\right(x)\nabla v)=\lambda \left(d\right(x)v+b(x\left)u\right)+F_v(x,u,v),x\in \Omega ,u{|}_{\partial \Omega }=v{|}_{\partial \Omega }=0$, where $\Omega \subset R^N(N\ge 2)$ is an open bounded domain with smooth boundary $\partial \Omega$, the measurable, nonnegative diffusion coefficients $h_1$, $h_2$ are allowed to vanish in $\Omega$ (as well as at the boundary $\partial \Omega$) and/or to blow up in $\overline{\Omega }$. Some multiplicity results of solutions are obtained for the degenerate elliptic systems which are near resonance at higher eigenvalues by the classical saddle point theorem and a local saddle point theorem in critical point theory.

We exhibit a regularity condition concerning the pressure gradient for the Navier-Stokes equations in a special class. It is shown that if the pressure gradient belongs to $L^{2/\mathrm{(2}-\mathrm{r)}}\left(\left(0,T\right);ℳ\left({\dot{H}}^r\left({ℝ}^3\right)\to {\dot{H}}^{-r}\left({ℝ}^3\right)\right)\right)$, where $ℳ\left({\dot{H}}^r\left({ℝ}^3\right)\to {\dot{H}}^{-r}\left({ℝ}^3\right)\right)$ is the multipliers between Sobolev spaces whose definition is given later for , then the Leray-Hopf weak solution to the Navier-Stokes equations is actually regular.

In (Wang et al., 2011), we give an iterative reproducing kernel method (IRKM). The main contribution of this paper is to use an IRKM (Wang et al., 2011), in singular perturbation problems with boundary layers. Two numerical examples are studied to demonstrate the accuracy of the present method. Results obtained by the method indicate that the method is simple and effective.

We consider the existence of the global attractor ${\mathcal{A}}_1$ for the 3D weakly damped wave equation. We prove that ${\mathcal{A}}_1$ is compact in $\left(H^2\right(\Omega )\cap H_0^1(\Omega \left)\right)\times H_0^1\left(\Omega \right)$ and attracts all bounded subsets of $\left(H^2\right(\Omega )\cap H_0^1(\Omega \left)\right)\times H_0^1\left(\Omega \right)$ with respect to the norm of $\left(H^2\right(\Omega )\cap H_0^1(\Omega \left)\right)\times H_0^1\left(\Omega \right)$. Furthermore, this attractor coincides with the global attractor in the weak energy space $H_0^1\left(\Omega \right)\times L^2\left(\Omega \right)$.

A nonlinear partial differential equation containing the famous Camassa-Holm and Degasperis-Procesi equations as special cases is investigated. The Kato theorem for abstract differential equations is applied to establish the local well-posedness of solutions for the equation in the Sobolev space $H^s\left(R\right)$ with $s>3/2$. Although the $H^1$-norm of the solutions to the nonlinear model does not remain constant, the existence of its weak solutions in the lower-order Sobolev space $H^s$ with $1\le s\le 3/2$ is proved under the assumptions $u_0\in H^s$ and .

Let $\mathbb{T}$ be a time scale and $a,b\in \mathbb{T}$, . We study the nonlinear fourth-order eigenvalue problem on $\mathbb{T}$, ${u^{\Delta }}^4\left(t\right)=\lambda h\left(t\right)f\left(u\right(t),{u^{\Delta }}^2(t\left)\right)$, $t\in [a,{\rho }^2(b){]}_{\mathbb{T}}$, $u\left(a\right)=u^{\Delta }\left(\sigma \right(b\left)\right)={u^{\Delta }}^2\left(a\right)={u^{\Delta }}^3\left(\rho \right(b\left)\right)=0$ and obtain the existence and nonexistence of positive solutions when and $\lambda >{\lambda }^{*}$, respectively, for some ${\lambda }^{*}$. The main tools to prove the existence results are the Schauder fixed point theorem and the upper and lower solution method.

The Cauchy problem for a nonlinear generalized pseudoparabolic equation is investigated. The well-posedness of local strong solutions for the problem is established in the Sobolev space $C\left(\right[0,T);H^s(R\left)\right){\bigcap }_{}{C}^1\left(\right[0,T);H^{s-1}(R\left)\right)$ with $s>3/2$, while the existence of local weak solutions is proved in the space $H^s\left(R\right)$ with $1\le s\le 3/2$. Further, under certain assumptions of the nonlinear terms in the equation, it is shown that there exists a unique global strong solution to the problem in the space $C\left(\right[0,\infty );H^s(R\left)\right){\bigcap }_{}{C}^1\left(\right[0,\infty );H^{s-1}(R\left)\right)$ with $s\ge 2$.

A shallow water wave equation with a weakly dissipative term, which includes the weakly dissipative Camassa-Holm and the weakly dissipative Degasperis-Procesi equations as special cases, is investigated. The sufficient conditions about the existence of the global strong solution are given. Provided that $(1-{\partial }_x^2)u_0\in M^{+}\left(R\right)$, $u_0\in H^1(R),$ and $u_0\in L^1\left(R\right)$, the existence and uniqueness of the global weak solution to the equation are shown to be true.

We discuss the existence of positive solutions of a boundary value problem of nonlinear fractional differential equation with changing sign nonlinearity. We first derive some properties of the associated Green function and then obtain some results on the existence of positive solutions by means of the Krasnoselskii's fixed point theorem in a cone.