We study the Cauchy problem for the 3D MHD system with damping terms ${\epsilon \left|u\right|}^{\alpha -1}u$ and ${\delta \left|b\right|}^{\beta -1}b$ (ε, δ > 0 and α, β ≥ 1), and show that the strong solution exists globally for any α, β > 3. This improves the previous results significantly.

The Cauchy problem for the system of linear generalized ordinary differential equations in the J. Kurzweil sense $\mathrm{d}x\left(t\right)=\mathrm{d}{A}_0\left(t\right)·x\left(t\right)+\mathrm{d}{f}_0\left(t\right)$, $x\left(t_0\right)=c_0$ $(t\in I)$ with a unique solution $x_0$ is considered. Necessary and sufficient conditions are obtained for a sequence of the Cauchy problems $\mathrm{d}x\left(t\right)=\mathrm{d}{A}_k\left(t\right)·x\left(t\right)+\mathrm{d}{f}_k\left(t\right)$, $x\left(t_k\right)=c_k$ $(k=1,2,\cdots )$ to have a unique solution $x_k$ for any sufficiently large $k$ such that $x_k\left(t\right)\to x_0\left(t\right)$ uniformly on $I$. Presented results are analogous to the sufficient conditions due to Z. Opial for linear ordinary differential systems....

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 the systems of hyperbolic equations ⎧$uₜₜ=\Delta \left(a(t,x)u\right)+\Delta \left(b(t,x)v\right)+{h(t,x)\left|v\right|}^p$, t > 0, $x\in {ℝ}^N$, (S1) ⎨ ⎩$vₜₜ=\Delta \left(c(t,x)v\right)+{k(t,x)\left|u\right|}^q$, t > 0, $x\in {ℝ}^N$ ⎧$uₜₜ=\Delta \left(a(t,x)u\right)+{h(t,x)\left|v\right|}^p$, t > 0, $x\in {ℝ}^N$, (S2) ⎨ ⎩$vₜₜ=\Delta \left(c(t,x)v\right)+{l(t,x)\left|v\right|}^m+k(t,x){\left|u\right|}^q$, t > 0, $x\in {ℝ}^N$, (S3) ⎧$uₜₜ=\Delta \left(a(t,x)u\right)+\Delta \left(b(t,x)v\right)+{h(t,x)\left|u\right|}^p$, t > 0, $x\in {ℝ}^N$, ⎨ ⎩$vₜₜ=\Delta \left(c(t,x)v\right)+{k(t,x)\left|v\right|}^q$, t > 0, $x\in {ℝ}^N$, in $(0,\infty )\times {ℝ}^N$ with u(0,x) = u₀(x), v(0,x) = v₀(x), uₜ(0,x) = u₁(x), vₜ(0,x) = v₁(x). We show that, in each case, there exists a bound B on N such that for 1 ≤ N ≤ B solutions to the systems blow up in finite time.

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...

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.

We study the chemotaxis system with singular sensitivity and logistic-type source: $u_t=\Delta u-\chi \nabla ·(u\nabla v/v)+ru-\mu u^k$, $0=\Delta v-v+u$ under the non-flux boundary conditions in a smooth bounded domain $\Omega \subset {ℝ}^n$, $\chi ,r,\mu >0$, $k>1$ and $n\ge 1$. It is shown with $k\in (1,2)$ that the system possesses a global generalized solution for $n\ge 2$ which is bounded when $\chi >0$ is suitably small related to $r>0$ and the initial datum is properly small, and a global bounded classical solution for $n=1$.

We also prove a long time existence result; more precisely we prove that for fixed $T>0$ there exists a set ${\Sigma }_T$, $\mathbb{P}\left({\Sigma }_T\right)>0$ such that any data ${\phi }^{\omega }\left(x\right)\in H^{\gamma }\left({𝕋}^3\right),\gamma <1,\omega \in {\Sigma }_T$, evolves up to time $T$ into a solution $u\left(t\right)$ with $u\left(t\right)-e^{it\Delta }{\phi }^{\omega }\in C([0,T];H^s\left({𝕋}^3\right))$, $s=s\left(\gamma \right)>1$. In particular we find a nontrivial set of data which gives rise to long time solutions below the critical space $H^1\left({𝕋}^3\right)$, that is in the supercritical scaling regime.

We show that every operator from ${\ell }_s$ to ${\ell }_p\otimes ̂{\ell }_q$ is compact when 1 ≤ p,q < s and that every operator from ${\ell }_s$ to ${\ell }_p\widehat{\otimes ̂}{\ell }_q$ is compact when 1/p + 1/q > 1 + 1/s.

Let $f^j={\sum }_k{a}_{k}f(2^{j+1}x-2k)$, where the sum is taken over the lattice of all points k in ${ℝ}^n$ having integer-valued components, j∈ℕ and $a_k\in ℂ$. Let $A_{pq}^s$ be either $B_{pq}^s$ or $F_{pq}^s$ (s ∈ ℝ, 0 < p < ∞, 0 < q ≤ ∞) on ${ℝ}^n.$ The aim of the paper is to clarify under what conditions $\parallel f^j|A_{pq}^s\parallel$ is equivalent to $2^{j(s-n/p)}({\sum }_k|a_k{|}^p{)}^{1/p}\parallel f|A_{pq}^s\parallel$.

In this work we study the problem $u_t{-div\left(\right|x|}^{-2\gamma }\nabla u)=\lambda \frac{u^{\alpha }}{{\left|x\right|}^{2(\gamma +1)}}+f$ in $\Omega \times (0,T)$, $u\ge 0$ in $\Omega \times (0,T)$, $u=0$ on $\partial \Omega \times (0,T)$, $u(x,0)=u_0\left(x\right)$ in $\Omega$, $\Omega \subset {ℝ}^N$ $(N\ge 2)$ is a bounded regular domain such that $0\in \Omega$, $\lambda >0$, $\alpha >0$, $-\infty <\gamma <\left(N-2\right)/2$, $f$ and $u_0$ are positive functions such that $f\in L^1(\Omega \times (0,T))$ and $u_0\in L^1\left(\Omega \right)$. The main points under analysis are: (i) spectral instantaneous and complete blow-up related to the Harnack inequality in the case $\alpha =1$, $1+\gamma >0$; (ii) the nonexistence of solutions if $\alpha >1$, $1+\gamma >0$; (iii) a uniqueness result for weak solutions (in the distribution sense); (iv) further results on existence of weak solutions...

We show that any ${\Sigma }_s$-product of at most $𝔠$-many $L\Sigma (\le \omega )$-spaces has the $L\Sigma (\le \omega )$-property. This result generalizes some known results about $L\Sigma (\le \omega )$-spaces. On the other hand, we prove that every ${\Sigma }_s$-product of monotonically monolithic spaces is monotonically monolithic, and in a similar form, we show that every ${\Sigma }_s$-product of Collins-Roscoe spaces has the Collins-Roscoe property. These results generalize some known results about the Collins-Roscoe spaces and answer some questions due to Tkachuk [Lifting the Collins-Roscoe...