Note on asymptotic behavior of weak solutions of the Cauchy problem for some parabolic equations with unbounded coefficients

1979 ◽  
Vol 121 (1) ◽  
pp. 207-215
Author(s):  
Lu-San Chen
1961 ◽  
Vol 13 ◽  
pp. 331-345 ◽  
Author(s):  
M. H. Protter

In this paper we shall be concerned with two problems: (i) the asymptotic behavior of solutions of parabolic inequalities and (ii) the uniqueness of the Cauchy problem for such inequalities when the data are prescribed on a portion of a time-like surface. The unifying feature of these rather separate problems is the employment of integral estimates of the same type in both cases.We consider parabolic operators in self-adjoint form(1)as well as the non-self-adjoint operator(2)where the coefficients aij(x, t) = aij (x1, x2... , xn, t) are C1 functions of x and t and the bij= bij(x, t) are C2 functions of x and t.


2018 ◽  
Vol 4 (2) ◽  
pp. 67-77
Author(s):  
Nicolau Matiel Lunardi Diehl ◽  
Lucinéia Fabris

In this paper, we show that the $L^1$ norm of the bounded weak solutions of the Cauchy problem for general degenerate parabolic equations of the formu_t + div f(x,t,u) = div(|u|^{\alpha}\nabla u),   x \in R^n , t > 0,where \alpha > 0 is constant, decrease, under fairly broad conditions in advection flow f. In addition, we derive the mass conservation property for positive (or negative) solutions.


1967 ◽  
Vol 29 ◽  
pp. 45-49 ◽  
Author(s):  
Tadashi Kuroda

Aronson proved, in his paper [1], the existence and the uniqueness property of weak solutions of the initial boundary value problem for parabolic equations of second order with measurable coefficients. On the uniqueness of solutions of the Cauchy problem for such equations he also gave some interesting results in [2].


2018 ◽  
Vol 4 (2) ◽  
pp. 67-77
Author(s):  
Nicolau Matiel Lunardi Diehl ◽  
Lucinéia Fabris

In this paper, we show that the $L^1$ norm of the bounded weak solutions of the Cauchy problem for general degenerate parabolic equations of the formu_t + div f(x,t,u) = div(|u|^{\alpha}\nabla u),   x \in R^n , t > 0,where \alpha > 0 is constant, decrease, under fairly broad conditions in advection flow f. In addition, we derive the mass conservation property for positive (or negative) solutions.


1971 ◽  
Vol 41 ◽  
pp. 33-42 ◽  
Author(s):  
Yoshiaki Ikeda

In this article we shall prove the uniqueness and existense of a weak solution for the Cauchy problem of linear parabolic equations with discontinuous and unbounded coefficients(1.1)


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