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Consider the equation $$ u'(t) = (Fu)(t) $$ where $F \colon L^2(0,T;\mathbb R^n) \to L^2(0,T;\mathbb R^n)$ is a causal (Volterra type) nonlinear operator. It means that the value of $(Fu)(t_0)$ depend …

Good day! Let $V = H^1(\Omega)$, $\Omega \subset \mathbb R^3$. Consider the space $W = \{ y \in L^2(0,T;V) \colon dy/dt \in L^2(0,T;V') \}$. It is well-known that $W \subset C([0,T];H)$ where $H = …

Good day! Let $V = H^1(\Omega)$, $\Omega \subset \mathbb R^3$. Consider the linear parabolic equation $y' + Ay = f$ where $f \in L^q(0,T;V')$, $y \in W = \{y \in L^p(0,T;V) \colon dy/dt \in L^q(0,T; …

Where can I find a theorem about Lyapunov stability for the equation in Hilbert space? $$ y' = Fy, $$ where $F : V \to V'$ is a nonlinear operator , $y' \in L^2(0,T,V')$, $V$ is a Hilbert space. Linea …

Good day! We have an equation $y'+Ay=Bu$ where $y=\{\theta,\varphi\}$, $A, B$ are nonlinear operators. $u \in L^\infty(\Gamma)$, $\theta, \varphi \in W = \{y \in L^2(0,T;V) : y'\in L^2(0,T;V')\}$, $ …