This paper provides blow up results of Fujita type for a reaction-diffusion system of 3 equations in the form $uₜ-\Delta \left(a_{11}u\right)=h(t,x){\left|v\right|}^p$, $vₜ-\Delta \left(a_{21}u\right)-\Delta \left(a_{22}v\right)=k(t,x){\left|w\right|}^q$, $wₜ-\Delta \left(a_{31}u\right)-\Delta \left(a_{32}v\right)-\Delta \left(a_{33}w\right)=l(t,x){\left|u\right|}^r$, for $x\in {ℝ}^N$, t > 0, p > 0, q > 0, r > 0, $a_{ij}=a_{ij}(t,x,u,v)$, under initial conditions u(0,x) = u₀(x), v(0,x) = v₀(x), w(0,x) = w₀(x) for $x\in {ℝ}^N$, where u₀, v₀, w₀ are nonnegative, continuous and bounded functions. Subject to conditions on dependence on the parameters p, q, r, N and the growth of the functions h, k, l at infinity, we prove finite blow up time for every solution of the...

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.

A Banach space $X$ has the reciprocal Dunford-Pettis property ($RDPP$) if every completely continuous operator $T$ from $X$ to any Banach space $Y$ is weakly compact. A Banach space $X$ has the $RDPP$ (resp. property $\left(wL\right)$) if every $L$-subset of $X^{*}$ is relatively weakly compact (resp. weakly precompact). We prove that the projective tensor product $X\otimes {}_{\pi }Y$ has property $\left(wL\right)$ when $X$ has the $RDPP$, $Y$ has property $\left(wL\right)$, and $L(X,Y^{*})=K(X,Y^{*})$.

In this paper we examine self-similar solutions to the system $u_{it}-d_i\Delta u_i={\prod }_{k=1}^m{u}_k^{p_k^i}$, i = 1,…,m, $x\in {ℝ}^N$, t > 0, $u_i(0,x)=u_{0i}\left(x\right)$, i = 1,…,m, $x\in {ℝ}^N$, where m > 1 and $p_k^i>0$, to describe asymptotics near the blow up point.

We consider problem (P) of Kirchhoff-Carrier type with nonlinear terms containing a finite number of unknown values $u({\eta }_1,t),\cdots ,u({\eta }_q,t)$ with $0\le {\eta }_1<{\eta }_2<\cdots <{\eta }_q<1.$ By applying the linearization method together with the Faedo-Galerkin method and the weak compact method, we first prove the existence and uniqueness of a local weak solution of problem (P). Next, we consider a specific case $\left({\mathrm{P}}_q\right)$ of (P) in which the nonlinear term contains the sum $S_q\left[u^2\right]\left(t\right)=q^{-1}{\sum }_{i=1}^q{u}^2(\frac{(i-1)}{q},t)$. Under suitable conditions, we prove that the solution of $\left({\mathrm{P}}_q\right)$ converges to the solution...

CONTENTSIntroduction............................................................................................................ 51. Preliminaries............................................................................................................. 82. Embedding into $W^{m,p}\left(\Omega \right)$ into $L^S\left(\Omega \right)$ (n>1).......................................... 103. The case n = 1.......................................................................................................... 284. Embedding $W^{m,p}\left(\Omega \right)$ into $L^{\phi }\left(\Omega \right)$...............................................................

Let $r(G_1,G_2)$ be the Ramsey number of the two graphs $G_1$ and $G_2$. For $n_1\ge n_2\ge 1$ let $S(n_1,n_2)$ be the double star given by $V\left(S(n_1,n_2)\right)=\{v_0,v_1,...,v_{n_1},w_0$, $w_1,...,w_{n_2}\}$ and $E\left(S(n_1,n_2)\right)=\{v_0{v}_1,...,v_0{v}_{n_1},v_0{w}_0,w_0{w}_1,...,w_0{w}_{n_2}\}$. We determine $r(K_{1,m-1},$ $S(n_1,n_2))$ under certain conditions. For $n\ge 6$ let $T_n^3=S(n-5,3)$, $T_n^{\text{'}\text{'}}=(V,E_2)$ and $T_n^{\text{'}\text{'}\text{'}}=(V,E_3)$, where $V=\{v_0,v_1,...,v_{n-1}\}$, $E_2=\{v_0{v}_1,...,v_0{v}_{n-4},v_1{v}_{n-3}$, $v_1{v}_{n-2},v_2{v}_{n-1}\}$ and $E_3=\{v_0{v}_1,...,v_0{v}_{n-4},v_1{v}_{n-3},$ $v_2{v}_{n-2},v_3{v}_{n-1}\}$. We also obtain explicit formulas for $r(K_{1,m-1},T_n)$, $r(T_m^{\text{'}},T_n)$ $(n\ge m+3)$, $r(T_n,T_n)$, $r(T_n^{\text{'}},T_n)$ and $r(P_n,T_n)$, where $T_n\in \{T_n^{\text{'}\text{'}},T_n^{\text{'}\text{'}\text{'}},T_n^3\}$, $P_n$ is the path on $n$ vertices and $T_n^{\text{'}}$ is the unique tree with $n$ vertices and maximal degree $n-2$.

We prove the existence of uniform attractors ${}_{\epsilon }$ in the space $H¹\left({ℝ}^N\right)$ for the nonautonomous nonclassical diffusion equation $u_t-\epsilon \Delta u_t-\Delta u+f(x,u)+\lambda u=g(x,t)$, ε ∈ [0,1]. The upper semicontinuity of the uniform attractors ${{}_{\epsilon }}_{\epsilon \in [0,1]}$ at ε = 0 is also studied.

CONTENTSIntroduction.......................................................................................61. Definition of certain special forms...........................................62. Statement of results...................................................................83. Proof of Theorem 2.....................................................................94. Preliminaries for Theorem 1.....................................................105. Further preliminaries for Theorem...

We study weakly precompact sets and operators. We show that an operator is weakly precompact if and only if its adjoint is pseudo weakly compact. We study Banach spaces with the $p$-$L$-limited${}^{*}$ and the $p$-(SR${}^{*}$) properties and characterize these classes of Banach spaces in terms of $p$-$L$-limited${}^{*}$ and $p$-Right${}^{*}$ subsets. The $p$-$L$-limited${}^{*}$ property is studied in some spaces of operators.

This paper is concerned with the study of a nonlocal nonlinear parabolic problem associated with the equation $u_t-M\left({\int }_{\Omega }\phi u\mathrm{d}x\right)\mathrm{div}\left(A(x,t,u)\nabla u\right)=g(x,t,u)$ in $\Omega \times (0,T)$, where $\Omega$ is a bounded domain of ${ℝ}^n$ $(n\ge 1)$, $T>0$ is a positive number, $A(x,t,u)$ is an $n\times n$ matrix of variable coefficients depending on $u$ and $M:ℝ\to ℝ$, $\phi :\Omega \to ℝ$, $g:\Omega \times (0,T)\times ℝ\to ℝ$ are given functions. We consider two different assumptions on $g$. The existence of a weak solution for this problem is proved using the Schauder fixed point theorem for each of these assumptions. Moreover, if $A(x,t,u)=a(x,t)$ depends only on...

Given an integer $n\ge 3$, let $u_1,...,u_n$ be pairwise coprime integers $\ge 2$, $𝒟$ a family of nonempty proper subsets of $\{1,...,n\}$ with “enough” elements, and $\epsilon$ a function $𝒟\to \{\pm 1\}$. Does there exist at least one prime $q$ such that $q$ divides ${\prod }_{i\in I}{u}_i-\epsilon \left(I\right)$ for some $I\in 𝒟$, but it does not divide $u_1\cdots u_n$? We answer this question in the positive when the $u_i$ are prime powers and $\epsilon$ and $𝒟$ are subjected to certain restrictions. We use the result to prove that, if ${\epsilon }_0\in \{\pm 1\}$ and $A$ is a set of three or more primes that contains all prime divisors of any...

We study nonlinear diffusion problems of the form $u_t=u_{xx}+f\left(u\right)$ with free boundaries. Such problems may be used to describe the spreading of a biological or chemical species, with the free boundary representing the expanding front. For special $f\left(u\right)$ of the Fisher-KPP type, the problem was investigated by Du and Lin [DL]. Here we consider much more general nonlinear terms. For any $f\left(u\right)$ which is $C^1$ and satisfies $f\left(0\right)=0$, we show that the omega limit set $\omega \left(u\right)$ of every bounded positive solution is determined by a stationary...

It is proved that real functions on $ℝ$ which can be represented as the difference of two semiconvex functions with a general modulus (or of two lower $C^1$-functions, or of two strongly paraconvex functions) coincide with semismooth functions on $ℝ$ (i.e. those locally Lipschitz functions on $ℝ$ for which $f_{+}^{\text{'}}\left(x\right)={lim}_{t\to x+}{f}_{+}^{\text{'}}\left(t\right)$ and $f_{-}^{\text{'}}\left(x\right)={lim}_{t\to x-}{f}_{-}^{\text{'}}\left(t\right)$ for each $x$). Further, for each modulus $\omega$, we characterize the class $DS{C}_{\omega }$ of functions on $ℝ$ which can be written as $f=g-h$, where $g$ and $h$ are semiconvex with modulus $C\omega$ (for some $C>0$) using a new...