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2011
It is established in Chap. 5 that the nonlinearity causes the blow-up to occur at a finite time in certain situations. If the solution to the ODE \(u_t \,= \,f(u)\), blows up at a finite time t = T with \(u(T - 0) = +\infty\), then u = G(T - t), where \(G(\xi)\) is the inverse function of \(\int\nolimits_\infty^u \frac {dn}{f(n)}\)
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It is established in Chap. 5 that the nonlinearity causes the blow-up to occur at a finite time in certain situations. If the solution to the ODE \(u_t \,= \,f(u)\), blows up at a finite time t = T with \(u(T - 0) = +\infty\), then u = G(T - t), where \(G(\xi)\) is the inverse function of \(\int\nolimits_\infty^u \frac {dn}{f(n)}\)
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Blow-up rates for parabolic systems
ZAMP Zeitschrift f�r angewandte Mathematik und Physik, 1996Two weakly coupled systems of parabolic equations are considered. One is coupled in the equations and the other in the boundary conditions. For both of them blow-up in finite time may occur. Estimates of the blow-up rates (in \(t\)) are established for certain classes of initial functions.
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Blow up rate for semilinear heat equations with subcritical nonlinearity
Indiana University Mathematics Journal, 2004The blow up rate of sign-changing solutions of the Cauchy problem for a semilinear heat equation is established under the assumption that the nonlinear source term is a subcritical power. This was known before for positive solutions, but for sign-changing solutions only a partial result was available [see \textit{Y. Giga} and \textit{R. V.
Giga, Y., Matsui, S., Sasayama, S.
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Mathematical Methods in the Applied Sciences, 2017
This paper deals with asymptotic behavior for blow‐up solutions to time‐weighted reaction–diffusion equations ut=Δu+eαtvp and vt=Δv+eβtuq, subject to homogeneous Dirichlet boundary. The time‐weighted blow‐up rates are defined and obtained by ways of the scaling or auxiliary‐function methods for all α, . Aiding by key inequalities between components of
Bingchen Liu, Fengjie Li
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This paper deals with asymptotic behavior for blow‐up solutions to time‐weighted reaction–diffusion equations ut=Δu+eαtvp and vt=Δv+eβtuq, subject to homogeneous Dirichlet boundary. The time‐weighted blow‐up rates are defined and obtained by ways of the scaling or auxiliary‐function methods for all α, . Aiding by key inequalities between components of
Bingchen Liu, Fengjie Li
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A Result on the Blow-up Rate for the Zakharov System in Dimension 3
SIAM Journal on Mathematical Analysis, 2001Summary: We consider a blow-up solution \((u,n,v)\) of the Zakharov system in \(\mathbb{R}^3\): \[ \begin{cases} iu_t= -\Delta u+nu,\\ n_t=- \nabla\cdot v,\\ v_t=-\nabla \bigl(n+|u|^2). \end{cases} \] If \(T\) is the finite blow-up time, we show the following integral estimate for \(n\): \[ \int^T_0 \left( \int_{\mathbb{R}^3}\bigl |n(x,t)\bigr |^q dx ...
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The blow‐up rate for the heat equation with a non‐linear boundary condition
Mathematical Methods in the Applied Sciences, 1991The authors consider the problem: \(v_ t=\Delta v\) in \(B\times(0,T)\); \(\partial v/\partial n=v^ p\) on \(\partial B\times(0,T)\); \(v(\zeta,0)=v_ 0(\zeta)\), \(\zeta\in\bar B\) where \(B:=\{\zeta\in\mathbb{R}^ n: |\zeta|1\) and \(v\geq0\) satisfies the boundary condition, is smooth, and has the form \(v_ 0(\zeta)=u_ 0(|\zeta|)\) for some \(u_ 0: [0,
Fila, Marek, Quittner, Pavol
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Determination of the blow-up rate for the semilinear wave equation
American Journal of Mathematics, 2003In this paper, we find the optimal blow-up rate for the semilinear wave equation with a power nonlinearity. The exponent p is superlinear and less than 1 + 4/ N -1 if N ≥ 2.
Merle, Franck, Hatem, Zaag
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Blow-up rates of large solutions for infinity Laplace equations
Applied Mathematics and Computation, 2017zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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Boundary blow‐up rate of solutions to elliptic cooperative systems
Mathematical Methods in the Applied Sciences, 2019AbstractThis paper shows the existence, uniqueness, and boundary blow‐up rate of large solution of cooperative systems of the form in a bounded smooth domain Ω⊂RN, bi(x) is nonnegative weight function that can be singular on ∂Ω, and the exponents verify λi∈R,ai>0,pi>1,qi>0 for i=1,2, and q1q2<p1p2.
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The Solutions for the Equations with Localized Consistent Blow-Up Rate
2011In this paper, The Dirichlet problem for a localized nonlinear equation ut1Â’1um = a(x)up(0,t)+b(x)uq(x,t) is investigated. In the case of p>q 1Â¥ m, It is proved that the solutions have a global blow-up and the rate of blow-up is uniform in all compact subsets of the domain.
Miaochao Chen, Jizhong Shi, Peishu Chen
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