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**Extra resources for Blow-Up in Quasilinear Parabolic Equations (De Gruyter Expositions in Mathematics)**

**Sample text**

Since Ilw(l, 1I1101l1-liRI An estimate of the effective width of the inhomogeneous temperature profile for large times is given by (21). In the next section we move on to analyze self-similar solutions of nonlinear heat equations. + 1/2 H I/2 Us(O)(1 + 7) 1/2: 7)(17. (20) Resolving consecutively all the indeterminacies that arise in the right-hand using the equality § 3 Asymptotic stability of self-similar solutions of nonlinear heat equations we obtain Let us consider flrst the example of a sclf-similar solution already encountered in Ch.

Since. as follows from (17), " IIw(I. 10 T)",- 112 + In this section we consider the Cauchy problem for the heat equation. Cd (\ + T)/11 J dT. II, :::;: U". t > O. x E (\) R. (28) o which is the same as (24). 11(0. x) :::;: 1I0(X) :::: O. x E R; 110 E (2) C(R). where the initial function has finite energy: Remark. The estimate (24) holds for sufficiently arbitrary (non-monotone in x) initial functions 110 E /}(R,). such that () ::: 11(1 ::: 110 ::: II;~ in R j , . where II~ are monotone functions, lI(f (0) :::;: III (0).

4) Here T > 0 is an arbitrary constant. Substituting this expression into ( I ) and taking into consideration the boundary condition, we obtain for Is :: 0 the following elliptic problem: (::"f~11 I + -Is = 0, x IT E U; I(x) = D, x E an. (5) For any IT > 0 it has a unique solution, strictly positive in n (existence of the solution can be established. for example, by constructing sub- and supersolutions of the problem; sec 17, 211) It turns out that (4) is stable with respect to arbitrary bounded perturbations of the initial function 1I0(X).