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1.

In this paper, we prove an existence result for \(\mathcal {L}^{\infty }\)-solutions for a class of semilinear delay evolution inclusions with measures and subjected to nonlocal initial conditions of the form

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle \mathrm{d}u(t)= \{Au(t)+f(t)\}\mathrm{d}t+\mathrm{d}h(t),&{}\quad t\in \mathbb {R}_+,\\ \displaystyle f(t)\in F(t,u_t),&{}\quad t\in \mathbb {R}_+,\\ \displaystyle u(t)=g(u)(t),&{}\quad t\in [\,-\tau ,0\,]. \end{array} \right. \end{aligned}$$

Here \(\tau \ge 0\), X is a Banach space, \(A:D(A)\subseteq X \rightarrow X \) is the infinitesimal generator of a \(C_0\)-semigroup, \(F:\mathbb {R}_+\times \mathcal {R}([\,-\tau ,0\,];X)\rightsquigarrow X\) is a u.s.c. multifunction with nonempty, convex and weakly compact values, \(h\in BV_{\mathrm{loc}}(\mathbb {R}_+;X)\) and the function \(g:\mathcal {R}_{b}(\mathbb {R}_+;X)\rightarrow \mathcal {R}([\,-\tau ,0\,];X)\) is nonexpansive.

  相似文献   

2.
Consider the following nonlinear singularly perturbed system of integral differential equations &\frac{\partial u}{\partial t}+f(u)+w\\ =&(\alpha-au)\int^{\infty}_0\xi(c)\left[\int_{\mathbb R}K(x-y) H\left(u\left(y,t-\frac1c|x-y|\right)-\theta\right){\rm d}y\right]{\rm d}c\\ &+(\beta-bu)\int^{\infty}_0\eta(\tau)\left[\int_{\mathbb R}W(x-y)H\big(u(y,t-\tau)-\Theta\big){\rm d}y\right]{\rm d}\tau,\\ &\frac{\partial w}{\partial t}=\varepsilon[g(u)-w], and the scalar integral differential equation &\frac{\partial u}{\partial t}+f(u)\\ =&(\alpha-a u)\int^{\infty}_0\xi(c)\left[\int_{\mathbb R}K(x-y) H\left(u\left(y,t-\frac1c|x-y|\right)-\theta\right){\rm d}y\right]{\rm d}c\\ &+(\beta-bu)\int^{\infty}_0\eta(\tau)\left[\int_{\mathbb R}W(x-y)H\big(u(y,t-\tau)-\Theta\big){\rm d}y\right]{\rm d}\tau. There exist standing wave solutions to the nonlinear system. Similarly, there exist standing wave solutions to the scalar equation. The author constructs Evans functions to establish stability of the standing wave solutions of the scalar equation and to establish bifurcations of the standing wave solutions of the nonlinear system.  相似文献   

3.
This paper deals with the existence of weak solutions to a class of degenerate and singular elliptic systems in ℝ N , N 2 of the form
$\left\{{l@{\quad}l}-\mathop{\mathrm{div}}(h_{1}(x)\nabla u)+a(x)u=f(x,u,v)&\mbox{in}\mathbb{R}^{N},\\-\mathop{\mathrm{div}}(h_{2}(x)\nabla v)+b(x)v=g(x,u,v)&\mbox{in}\mathbb{R}^{N},\right.$\left\{\begin{array}{l@{\quad}l}-\mathop{\mathrm{div}}(h_{1}(x)\nabla u)+a(x)u=f(x,u,v)&\mbox{in}\mathbb{R}^{N},\\-\mathop{\mathrm{div}}(h_{2}(x)\nabla v)+b(x)v=g(x,u,v)&\mbox{in}\mathbb{R}^{N},\end{array}\right.  相似文献   

4.
利用变分原理和Z2不变群指标研究了二阶常微分方程边值问题{u″(t)-u(t) f(t,u(t))=0,0<t<1,u′(0)=0,α1u(1) u′(1)=0,(其中α1>-1/2).得出了这类方程存在无穷个解的充分条件.  相似文献   

5.
1.IntroductionNeutraldelayffereatialequationsinpopulationdynamicshavebeenstudiedextensivelyforthep88tfewyears.However,onlyafewpapersll--5]havebeenpublishedontheexistencesofperiodicsolutionsoftheneutraldelaypopulationmodels.In[6,7],Kuangproposedtoinve...  相似文献   

6.
In the present paper, we deal with the existence and multiplicity of solutions for the following impulsive fractional boundary value problem
$$\begin{aligned} {_{t}}D_{T}^{\alpha }\left( \left| {_{0}}D_{t}^{\alpha }u(t)\right| ^{p-2}{_{0}}D_{t}^{\alpha }u(t)\right) + a(t)|u(t)|^{p-2}u(t)= & {} f(t,u(t)),\;\;t\ne t_j,\;\;\hbox {a.e.}\;\;t\in [0,T],\\ \Delta \left( {_{t}}I_{T}^{1-\alpha }\left( \left| {_{0}}D_{t}^{\alpha }u(t_j)\right| ^{p-2}{_{0}}D_{t}^{\alpha }u(t_j)\right) \right)= & {} I_j(u(t_j))\;\;j=1,2,\ldots ,n,\\ u(0)= & {} u(T) = 0. \end{aligned}$$
where \(\alpha \in (1/p, 1]\), \(1<p<\infty \), \(0 = t_0<t_1< t_2< \cdots< t_n < t_{n+1} = T\), \(f:[0,T]\times \mathbb {R} \rightarrow \mathbb {R}\) and \(I_j : \mathbb {R} \rightarrow \mathbb {R}\), \(j = 1, \ldots , n\), are continuous functions, \(a\in C[0,T]\) and
$$\begin{aligned} \Delta \left( {_{t}}I_{T}^{1-\alpha }\left( \left| {_{0}}D_{t}^{\alpha }u(t_j)\right| ^{p-2}{_{0}}D_{t}^{\alpha }u(t_j)\right) \right)= & {} {_{t}}I_{T}^{1-\alpha }\left( \left| {_{0}}D_{t}^{\alpha }u\left( t_j^+\right) \right| ^{p-2}{_{0}}D_{t}^{\alpha }u\left( t_j^+\right) \right) \\&- {_{t}}I_{T}^{1-\alpha }\left( \left| {_{0}}D_{t}^{\alpha }u(t_j^-)\right| ^{p-2}{_{0}}D_{t}^{\alpha }u\left( t_j^-\right) \right) ,\\ {_{t}}I_{T}^{1-\alpha }\left( \left| {_{0}}D_{t}^{\alpha }u\left( t_j^+\right) \right| ^{p-2}{_{0}}D_{t}^{\alpha }u\left( t_j^+\right) \right)= & {} \lim _{t \rightarrow t_j^+} {_{t}}I_{T}^{1-\alpha }\left( \left| {_{0}}D_{t}^{\alpha }u(t)\right| ^{p-2}{_{0}}D_{t}^{\alpha }u(t)\right) ,\\ {_{t}}I_{T}^{1-\alpha }\left( \left| {_{0}}D_{t}^{\alpha }u(t_j^-)\right| ^{p-2}{_{0}}D_{t}^{\alpha }u(t_j^-)\right)= & {} \lim _{t\rightarrow t_j^-}{_{t}}I_{T}^{1-\alpha }\left( \left| {_{0}}D_{t}^{\alpha }u(t)\right| ^{p-2}{_{0}}D_{t}^{\alpha }u(t)\right) . \end{aligned}$$
By using variational methods and critical point theory, we give some criteria to guarantee that the above-mentioned impulsive problems have at least one weak solution and a sequences of weak solutions.
  相似文献   

7.
Fujita exponents for evolution problems with nonlocal diffusion   总被引:1,自引:0,他引:1  
We prove the existence of a critical exponent of Fujita type for the nonlocal diffusion problem
$\left\{{l@{\quad}l}u_t(x, t) = J*u(x, t)-u(x, t) + u^p(x, t), & \qquad x \in \mathbb{R}^N,\; t > 0,\\ u(x, 0) = u_0(x), & \qquad x \in\mathbb{R}^N,\right.$\left\{\begin{array}{l@{\quad}l}u_t(x, t) = J*u(x, t)-u(x, t) + u^p(x, t), & \qquad x \in \mathbb{R}^N,\; t > 0,\\ u(x, 0) = u_0(x), & \qquad x \in\mathbb{R}^N,\end{array}\right.  相似文献   

8.
This paper deals with a two-competing-species chemotaxis system with two different chemicals
$$\begin{aligned} \left \{ \textstyle\begin{array}{l@{\quad}l} \displaystyle u_{t}=\Delta u-\chi_{1}\nabla \cdot (u\nabla v)+\mu_{1} u(1-u-a _{1}w), & (x,t)\in \varOmega \times (0,\infty ), \\ \displaystyle \tau v_{t}=\Delta v-v+w, & (x,t)\in \varOmega \times (0,\infty ), \\ \displaystyle w_{t}=\Delta w-\chi_{2}\nabla \cdot (w\nabla z)+\mu_{2}w(1-a_{2}u-w), & (x,t)\in \varOmega \times (0,\infty ), \\ \displaystyle \tau z_{t}=\Delta z-z+u, & (x,t)\in \varOmega \times (0,\infty ), \end{array}\displaystyle \right . \end{aligned}$$
under homogeneous Neumann boundary conditions in a smooth bounded domain \(\varOmega \subset \mathbb{R}^{n}\) \((n\geq 1)\) with the nonnegative initial data \((u_{0},\tau v_{0},w_{0},\tau z_{0})\in C^{0}(\overline{\varOmega }) \times W^{1,\infty }(\varOmega )\times C^{0}(\overline{\varOmega })\times W ^{1,\infty }(\varOmega )\), where \(\tau \in \{0,1\}\) and the parameters \(\chi_{i},\mu_{i},a_{i}\) (\(i=1,2\)) are positive. When \(\tau =0\), based on some a priori estimates and Moser-Alikakos iteration, it is shown that regardless of the size of initial data, the system possesses a unique globally bounded classical solution for any positive parameters if \(n=2\). On the other hand, when \(\tau =1\), relying on the maximal Sobolev regularity and semigroup technique, it is proved that the system admits a unique globally bounded classical solution provided that \(n\geq 1\) and there exists \(\theta_{0}>0\) such that \(\frac{\chi_{2}}{ \mu_{1}}<\theta_{0}\) and \(\frac{\chi_{1}}{\mu_{2}}<\theta_{0}\).
  相似文献   

9.
Results on finite-time blow-up of solutions to the nonlocal parabolic problem

are established. They extend some known results to higher dimensions.

  相似文献   


10.
In this paper we deal with local estimates for parabolic problems in ${\mathbb{R}^N}$ with absorbing first order terms, whose model is $$\left\{\begin{array}{l@{\quad}l}u_t- \Delta u +u |\nabla u|^q = f(t,x) \quad &{\rm in}\, (0,T) \times \mathbb{R}^N\,,\\u(0,x)= u_0 (x) &{\rm in}\, \mathbb{R}^N \,,\quad\end{array}\right.$$ where ${T >0 , \, N\geq 2,\, 1 < q \leq 2,\, f(t,x)\in L^1\left( 0,T; L^1_{\rm loc} \left(\mathbb{R}^N\right)\right)}$ and ${u_0\in L^1_{\rm loc}\left(\mathbb{R}^{N}\right)}$ .  相似文献   

11.
Let Ω be an open, bounded domain in \mathbbRn  (n ? \mathbbN){\mathbb{R}^n\;(n \in \mathbb{N})} with smooth boundary ∂Ω. Let p, q, r, d 1, τ be positive real numbers and s be a non-negative number which satisfies 0 < \fracp-1r < \fracqs+1{0 < \frac{p-1}{r} < \frac{q}{s+1}}. We consider the shadow system of the well-known Gierer–Meinhardt system:
$ \left \{ {l@{\quad}l} \displaystyle{u_t = d_1\Delta u - u + \frac{u^p}{\xi^q}}, & \quad {\rm in}\;\Omega \times (0,T), \\ \displaystyle{\tau \xi_t = -\xi + \frac{1}{|\Omega|} \int\nolimits_\Omega\frac{u^r}{\xi^s} {\rm d}x}, & \quad {\rm in}\;(0,T), \\ \displaystyle{\frac{\partial u}{\partial \nu} =0}, & \quad {\rm on}\;\partial \Omega \times (0,T), \\ \displaystyle{\xi(0) = \xi_0 >0 , \quad u(\cdot,0) = u_0(\cdot)} \geq 0 & \quad {\rm in}\;\Omega. \right. $ \left \{ \begin{array}{l@{\quad}l} \displaystyle{u_t = d_1\Delta u - u + \frac{u^p}{\xi^q}}, & \quad {\rm in}\;\Omega \times (0,T), \\ \displaystyle{\tau \xi_t = -\xi + \frac{1}{|\Omega|} \int\nolimits_\Omega\frac{u^r}{\xi^s} {\rm d}x}, & \quad {\rm in}\;(0,T), \\ \displaystyle{\frac{\partial u}{\partial \nu} =0}, & \quad {\rm on}\;\partial \Omega \times (0,T), \\ \displaystyle{\xi(0) = \xi_0 >0 , \quad u(\cdot,0) = u_0(\cdot)} \geq 0 & \quad {\rm in}\;\Omega. \end{array} \right.  相似文献   

12.
In this paper we deal with the existence of positive solutions for the following nonlocal type of problems $$\everymath{\displaystyle} \left\{ \begin{array}{l@{\quad}l} -\Delta u = \frac{\sigma}{( \int_{\varOmega} g(u)\, dx )^p} f(u) & \mbox{in}\ \varOmega, \\[3mm] u>0 & \mbox{in}\ \varOmega, \\[1mm] u=0 & \mbox{on}\ \partial\varOmega, \end{array} \right. $$ where Ω is a bounded smooth domain in ? N (N≥1), f,g are continuous positive functions, σ>0 and p∈?. We give sufficient conditions on the functions f and g in order to have existence of positive solutions.  相似文献   

13.
In this paper, we study the existence of infinitely many high energy solutions for the nonlinear Kirchhoff equations $$\left\{\everymath{\displaystyle}\begin{array}{l@{\quad}l}- \biggl(a+b\int_{R^3} |\nabla u|^2 dx\biggr)\Delta u + V(x)u=f(x,u),&x\in \mathbb {R}^3,\\[9pt]u\in H^1 (\mathbb {R}^3),\end{array}\right.$$ where a,b>0 are constants, V:?3→? is continuous and has a positive infimum. f is a subcritical nonlinearity which needs not to satisfy the usual Ambrosetti-Rabinowitz-type growth conditions.  相似文献   

14.
We provide two regularity criteria for the weak solutions of the 3D micropolar fluid equations, the first one in terms of one directional derivative of the velocity, i.e., $\partial_{3}u$, while the second one is is in terms of the behavior of the direction of the velocity $\frac{u}{|u|}$. More precisely, we prove that if \begin{equation*} \partial_{3}u \in L^{\beta}(0,T;L^{\alpha}(\mathbb{R}^{3}))\quad\text{ with }\frac{2}{\beta}+\frac{3}{\alpha}\leq 1+\frac{1}{\alpha}, 2&lt; \alpha \leq\infty, 2\leq\beta&lt; \infty; \end{equation*} or \begin{equation*} \operatorname{div}\left(\frac{u}{|u|}\right)\in L^{\frac{4}{1-2r}}(0,T;\dot{X}_{r}(\mathbb{R}^{3}))\quad \text{ with } 0\leq r&lt; \frac{1}{2}, \end{equation*} then the weak solution $(u(x,t),\omega(x,t))$ is regular on $\mathbb{R}^{3}\times [0,T]$. Here $\dot{X}_{r}(\mathbb{R}^{3})$ is the multiplier space.  相似文献   

15.
This paper is concerned with the following Kirchhoff-type equation
$$\begin{aligned} -\left( a+b\int _{\mathbb {R}^3}|\nabla {u}|^2\mathrm {d}x\right) \triangle u+V(x)u=f(x, u), \quad x\in \mathbb {R}^{3}, \end{aligned}$$
where \(V\in \mathcal {C}(\mathbb {R}^{3}, (0,\infty ))\), \(f\in \mathcal {C}({\mathbb {R}}^{3}\times \mathbb {R}, \mathbb {R})\), V(x) and f(xt) are periodic or asymptotically periodic in x. Using weaker assumptions \(\lim _{|t|\rightarrow \infty }\frac{\int _0^tf(x, s)\mathrm {d}s}{|t|^3}=\infty \) uniformly in \(x\in \mathbb {R}^3\) and
$$\begin{aligned}&\left[ \frac{f(x,\tau )}{\tau ^3}-\frac{f(x,t\tau )}{(t\tau )^3} \right] \mathrm {sign}(1-t) +\theta _0V(x)\frac{|1-t^2|}{(t\tau )^2}\ge 0, \quad \\&\quad \forall x\in \mathbb {R}^3,\ t>0, \ \tau \ne 0 \end{aligned}$$
with a constant \(\theta _0\in (0,1)\), instead of the common assumption \(\lim _{|t|\rightarrow \infty }\frac{\int _0^tf(x, s)\mathrm {d}s}{|t|^4}=\infty \) uniformly in \(x\in \mathbb {R}^3\) and the usual Nehari-type monotonic condition on \(f(x,t)/|t|^3\), we establish the existence of Nehari-type ground state solutions of the above problem, which generalizes and improves the recent results of Qin et al. (Comput Math Appl 71:1524–1536, 2016) and Zhang and Zhang (J Math Anal Appl 423:1671–1692, 2015). In particular, our results unify asymptotically cubic and super-cubic nonlinearities.
  相似文献   

16.
The solution u of the well-posed problem
depends continuously on (a ij ,β,γ,q). Dedicated to Karl H. Hofmann on his 75th birthday.  相似文献   

17.
We consider the following nonperiodic diffusion systems
$ \left\{{ll} \partial_{t}u-\triangle_{x}u+b(t,x)\nabla_{x}u+V(x)u=G_{v} (t,x,u,v), \\ -\partial_{t}v-\triangle_{x}v-b(t,x)\nabla_{x}v+V(x)v=G_{u} (t,x,u,v), \right. {\forall}(t,x)\in\mathbb{R} \times\mathbb{R}^{N}, $ \left\{\begin{array}{ll} \partial_{t}u-\triangle_{x}u+b(t,x)\nabla_{x}u+V(x)u=G_{v} (t,x,u,v), \\ -\partial_{t}v-\triangle_{x}v-b(t,x)\nabla_{x}v+V(x)v=G_{u} (t,x,u,v), \end{array}\right. {\forall}(t,x)\in\mathbb{R} \times\mathbb{R}^{N},  相似文献   

18.
Let T 1 be an integer, T = {0, 1, 2,..., T- 1}. This paper is concerned with the existence of periodic solutions of the discrete first-order periodic boundary value problems△u(t)- a(t)u(t) = λu(t) + f(u(t- τ(t)))- h(t), t ∈ T,u(0) = u(T),where △u(t) = u(t + 1)- u(t), a : T → R and satisfies∏T-1t=0(1 + a(t)) = 1, τ : T → Z t- τ(t) ∈ T for t ∈ T, f : R → R is continuous and satisfies Landesman-Lazer type condition and h : T → R. The proofs of our main results are based on the Rabinowitz's global bifurcation theorem and Leray-Schauder degree.  相似文献   

19.
In this paper, we consider the viscoelastic wave equation with a delay term in internal feedbacks; namely, we investigate the following problem
(x,t)- u(x,t)+_0^tg(t-s)u(x,s)ds+_1u_t(x,t)+_2 u_t(x,t-)=0u_{tt}(x,t)-\Delta u(x,t)+\int\limits_{0}^{t}g(t-s){\Delta}u(x,s){d}s+\mu_{1}u_{t}(x,t)+\mu_{2} u_{t}(x,t-\tau)=0  相似文献   

20.
In this paper, we study the following delayed predator-prey model of prey dispersal in two-patch environments $$\begin{array}{rcl}\dot{x}_1(t)&=&\displaystyle x_1(t)[r_1(t)-a_{11}(t)x_1(t)-a_{13}(t)y(t)]+D(t)(x_2(t)-x_1(t)),\\[3mm]\dot{x}_2(t)&=&\displaystyle x_2(t)[r_2(t)-a_{22}(t)x_2(t)-a_{23}(t)y(t)]+D(t)(x_1(t)-x_2(t)),\\[3mm]\dot{y}(t)&=&\displaystyle y(t)[-r_3(t)+a_{31}(t)x_1(t-\tau_1)+a_{32}(t)x_2(t-\tau _1)-a_{33}(t)y(t-\tau_2)].\end{array}$$ By giving the detail analyzing of the right-hand side functional of the system, sufficient and necessary condition which guarantee the predator and the prey species to be permanent are obtained. Numeric simulations show the feasibility of main results. In additional to the above, sufficient condition on the permanence of the above system with predator density-independence are established.  相似文献   

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