A one-dimensional nonlinear heat equation with a singular term

作者：Zhou, Wenshu^*; Lei, Peidong

来源：Journal of Mathematical Analysis and Applications, 2010, 368(2): 711-726.

DOI：10.1016/j.jmaa.2010.03.066

• Heat equation
• Weak solution
• Nonuniqueness
• Stationary solution

In this paper we are concerned with the Dirichlet problem for the one-dimensional nonlinear heat equation with a singular term:
{u(t) = u(xx) - sigma u(m)u(x)(2) + f(x, t), u > 0, (x, t) is an element of Q(T), u(a, t) = u(b, t) = 0, t is an element of [0, T], u(x, 0) = u(0)(x), x is an element of l,
where T > 0, Q(T) = 1 x (0, T], I = (a, b) with a < b, sigma > 0, -1 >= m > -2. We find that the problem may have multiple weak solutions for some initial data. To prove this, we need to study existence of positive classical solutions. In addition, we also discuss existence of a positive stationary solution for the above problem and relations between solutions of the above problem and the following problem: <br>

• 出版日期2010-8-15
• 单位大连民族大学; 东北师范大学