共查询到20条相似文献,搜索用时 15 毫秒
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We consider a nonlinear parabolic equation containing the porous medium operator and a nonlinear absorption term, which causes
the appearance of a moving boundary. Basic behavior and regularity results are obtained for the solution and the moving boundary
under two different boundary conditions. Also, the behavior of the solution and the moving boundary as time goes to infinity
is investigated.
Supported in part by NSF grant number MCS 8104220. 相似文献
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Eurica Henriques 《Journal of Differential Equations》2008,244(10):2578-2601
We extend to the singular case the results of [E. Henriques, J.M. Urbano, Intrinsic scaling for PDEs with an exponential nonlinearity, Indiana Univ. Math. J. 55 (5) (2006) 1701-1721] concerning the regularity of weak solutions of the porous medium equation with variable exponent. The method of intrinsic scaling is used to show that local weak solutions are locally continuous. 相似文献
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This paper studies the Cauchy problem for a doubly nonlinear parabolic equation. The main result shows that if there is a nonnegative solution of the Cauchy problem, then the initial trace of the solution is uniquely given as a nonnegative Borel measure satisfying an exponential growth condition. This extends the known result for the heat equation to the nonlinear case. 相似文献
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We consider a system of two porous medium equations defined on two different components of the real line, which are connected by the nonlinear contact condition
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In this paper we analyze the large time behavior of nonnegative solutions of the Cauchy problem of the porous medium equation with absorption u t-u m + γup = 0, where γ≥ 0, m 1 and p m + 2/N . We will show that if γ = 0 and 0 μ 2N/(N(m-1)+2), or γ 0 and 1/(p-1)μ2N/(N(m-1)+2), then for any nonnegative function φ in a nonnegative countable subset F of the Schwartz space S (R N ), there exists an initial-value u0 ∈C(RN) with lim x →∞ u 0 (x) = 0 such that φ is an ω-limit point of the rescaled solutions t μ/2 u(t β·, t), where β =[2-μ(m-1)]/4 . 相似文献
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Kin Ming Hui 《Transactions of the American Mathematical Society》1998,350(11):4651-4667
We prove that as the solutions of , , , , , , , converges in to the solution of the ODE , , where , , satisfies in for some function , , satisfying whenever for a.e. , for and for , where is a constant and is any measurable subset of .
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Multiple blow-up for a porous medium equation with reaction 总被引:1,自引:0,他引:1
The present paper is concerned with the Cauchy problem
$\left\{{ll}\partial_t u = \Delta u^m + u^p & \quad {\rm in}\; \mathbb R^N \times (0,\infty),\\ u(x,0) = u_0(x) \geq 0 & \quad {\rm in}\; \mathbb R^N, \right.$\left\{\begin{array}{ll}\partial_t u = \Delta u^m + u^p & \quad {\rm in}\; \mathbb R^N \times (0,\infty),\\ u(x,0) = u_0(x) \geq 0 & \quad {\rm in}\; \mathbb R^N, \end{array}\right. 相似文献
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Teemu Lukkari 《Journal of Evolution Equations》2010,10(3):711-729
We study the existence of solutions to the porous medium equation with a nonnegative, finite Radon measure on the right-hand
side. We show that such problems have solutions in a wide class of supersolutions. These supersolutions are defined as lower
semicontinuous functions obeying a parabolic comparison principle with respect to continuous solutions. We also consider the
question of how the integrability of the gradient of solutions is affected if the measure is given by a function in L
s
, for a small exponent s > 1. 相似文献
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《Communications in Nonlinear Science & Numerical Simulation》2014,19(2):371-376
In Ibragimov (2007) [13] a general theorem on conservation laws was proved. In Gandarias (2011) and Ibragimov (2011) [7], [15] the concepts of self-adjoint and quasi self-adjoint equations were generalized and the definitions of weak self-adjoint equations and nonlinearly self-adjoint equations were introduced. In this paper, we find the subclasses of nonlinearly self-adjoint porous medium equations. By using the property of nonlinear self-adjointness, we construct some conservation laws associated with classical and nonclassical generators of the differential equation. 相似文献
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M.L. Gandarias 《Communications in Nonlinear Science & Numerical Simulation》2012,17(6):2342-2349
The concepts of self-adjoint and quasi self-adjoint equations were introduced by Ibragimov (2006, 2007) [4], [7]. In Ibragimov (2007) [6] a general theorem on conservation laws was proved. In Gandarias (2011) [3] we generalized the concept of self-adjoint and quasi self-adjoint equations by introducing the definition of weak self-adjoint equations. In this paper we find the subclasses of weak self-adjoint porous medium equations. By using the property of weak self-adjointness we construct some conservation laws associated with symmetries of the differential equation. 相似文献
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Changfeng Gui Xiaosong Kang 《Transactions of the American Mathematical Society》2004,356(11):4273-4285
We prove the strict localization for a porous medium type equation with a source term, , , 1$">, \sigma +1$">, 0,$"> in the case of arbitrary compactly supported initial functions . We also otain an estimate of the size of the localization in terms of the support of the initial data and the blow-up time . Our results extend the well-known one dimensional result of Galaktionov and solve an open question regarding high dimensions.
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In this paper we study the large time behavior of non-negative solutions to the Cauchy problem of ut=Δum−uq in RN×(0,∞), where m>1 and q=qc≡m+2/N is a critical exponent. For non-negative initial value u(x,0)=u0(x)∈L1(RN), we show that the solution converges, if u0(x)(1+|x|)k is bounded for some k>N, to a unique fundamental solution of ut=Δum, independent of the initial value, with additional logarithmic anomalous decay exponent in time as t→∞. 相似文献
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