A new kind of the solution of degenerate parabolic equation with unbounded convection term

Abstract A new kind of entropy solution of Cauchy problem of the strong degenerate parabolic equation is introduced. If u0 ∈ L∞(RN), E = {Ei} ∈ (L2(QT))N and divE ∈ L2(QT), by a modified regularization method and choosing the suitable test functions, the BV estimates are got, the existence of the entropy solution is obtained. At last, by Kruzkov bi-variables method, the stability of the solutions is obtained.


Introduction
Consider the equation @u @t D @ @x i .a.u; x; t / @u @x i / C div.E.x; t /u/; .x; t / 2 Q D .0; T /: where is a open domain in R N , Assuming that 0 <˛Ä a.x; t; s/ Äˇ, some applicative models related to equation (1) were studied in [12]. If is bounded, Boccardo L., Orsina L. and Porretta A. in [2] defined that: u 2 L 1 .Q/ T L 2 .0; T I H 1 0 . // is a weak solution of equation (1) in the sense of that < u t ; ' > C " Q a.u; x; t /ru r'dxdt D " Q uEr'dxdt; (2) for every ' 2 L 2 .0; T I H 1 0 . //, where < :; : > denotes the duality product between L 2 .0; T I H 1 0 . // and L 2 .0; T I H 1 . //. Clearly, if a.x; t; s/ Á 0, then (1) becomes the type of conservation law equation, and it is well known that in this case, if one defines the weak solution as (2), then the uniqueness of the solutions is not true.
Also, Boccardo L., Orsina L. and Porretta A. in [2] had introduced the following unbounded entropy solution.
for almost every t 2 .0; T /, for every ' 2 L 2 .0; T I H 1 0 . // T L 1 .Q/ such that ' t 2 L 2 .0; T I H 1 . // C L 1 .Q/, where ‚ k .s/ D R s 0 T k .r/dr. If we check the proof of the theorems in [2], we have found that the condition 0 <˛Ä a.x; t; s/ Äˇacts an important role. If this condition is weakened to 0 Ä˛, to get the same conclusions seems difficult. By the way, though the authors did not discussed the uniqueness of the solutions in [2], we believe that the uniqueness of the solutions defined in the sense of inequality (3) is true, and we may study this problem in the future.
In our paper, we will consider Cauchy problem u t div.a.u; x; t /ru/ D div.uE/; .x; t / 2 Q T D R N .0; T /; (4) u.
Comparing equation (6) with equation (4), the main obstacle is the unboundedness of E. The unboundedness of E makes the estimating method used in [3,8,14,23,25] et al. not effective. To overcome these difficulties, we put forward a new definition of BV-entropy solution for problem (4)- (5), and by modifying the classical parabolic regularizing method, we are able to get the BV estimated formulas. The method used in our paper is completely different from that used in [18,20,24], [3,8,14,23,25] et al. But we use some inspiring techniques in [21]. To the end, some restrictions in E are added.

Definition and main results
where r D p a , and O r.u; x; t / D 1 Z 0 q a.su C C .1 s/u ; x; t /ds: (2) For any ' 2 C 2 0 .Q T /; ' 0; k 2 R; Á > 0; u satisfies where the pairs of equal indices imply a summation from 1 up to N , and Clearly if u is the a solution in Definition 2.1, then u is the entropy solution defined in [18].
The main results of the paper are the following theorems.
.a x s / 2 0; then problem (4)-(5) has a generalized solution in the sense of Definition 2.1, where x N C1 D t.
respectively. Suppose that A.s; x; t / satisfies the conditions as in Theorem 2.2, and where c; are positive constants and Corollary 2.4. The solution of problem (4)-(5) is unique.
To explain the reasonableness of Definition 2.1, suppose that equation (4) has a classical solution u. Let ' 2 C 2 0 .Q T /; ' 0; k 2 R; Á > 0. Multiplying (4) by 'S Á .u k/ and integrating over Q T , we have " Then " where So " where g i D p a @u @x i .
By (14)- (19), if equation (4) has a classical solution u, then Clearly Let Á ! 0 in this inequality. We have Clearly, if one defines the weak solutions u 1 ; u 2 , and u 3 of equation (4) (similarly, also equation (6)) in the sense of formulas (8), (21) and (8') respectively, then u 1 is also a solution in the senses of inequality (21) and (8'), u 2 is also the solution in the sense of inequality (8'). If equation (6) is weakly degenerate, Ref. [11,15,17,25] adopted to define the weak solution in sense (8'). In this case, the term S 0 (20) seems redundant, and should be drawn away. But, if equation (6) is strongly degenerate, the references [3,4,8,14,23,25] tell us that the term S 0 Á .u k/ P N j D1 j g j j 2 ' implies very important information of the uniqueness, it can not be drawn away.
Also, we note that the classical solution u induces an integral equality (20), while the weak solution formula defined as (8) is an inequality, this is due to that the following weak convergence property.
Generally, inequality (22) can not be an equality. In what follows, one can see that this is why we can only define the weak solution as (8) instead of (20).
Remark 2.6. Consider the equation Vol 0 pert A.I. and Hudjaev S.I. in [21] defined that: We know that only under the condition @A.u/ @x 2 L 1 Q T / T BV x .Q T / the uniqueness of the solutions in the sense (23) is true. So, an essential improvement of our paper (also [3,4,8,14,23,25]) is to get the uniqueness of the solutions in the sense (8) without any bounded restrictions in a.

The regularized problem and the proof of Theorem 2.2
Suppose that A.s; x; t /; u 0 .x/ are smooth as in the assumption of Theorem 2.2 and u 0 .
For any given large positive numbers K, let us introduce the following modified regularized equation.
where ı " is the mollifier as usual, i.e. let y D .x; t / D .x 1 ; ; x N ; t /, and For any given " > 0, let Here, we choose " D 1 K especially, and Moreover, we suppose that suppu 0K B K D fx W jxj < Kg, and it satisfies It is well-known that there is a classical solutions u K 2 C 2;1 .Q T / of (25)-(26). By this fact and using the maximum principle, we have Let gradu K D .u x 1 ; u x 2 ; ; u x N ; u x N C1 / and u x N C1 D u t . For simplicity, we denote u K as u in the following calculation. Let us derivation on x s ; s D 1; 2; ; N; N C 1 in (25). Then multiplying with u x s S Á .jgraduj/ jgraduj ' on the two sides, 0 Ä ' 2 C 1 0 .Q T /, and integrating over B K , we get For the last term of the left hand side in (29), For the other terms of the left hand side in (29), integrating by part, we have If we notice that " D 1 K , then so, by the facts of that jK.x y/j < 1, jK.x y/j 2 1 dy Ä cK N C3 : Thus, if we choose that Similarly, we are able to show that j @ 2 .ı " T K .E// @x s @x i j Ä cK N C4 ; At the same time, if we set q 12 q 1N C1 : : : where .q sp / is the square root of By the assumption By a process of limit, one can assume that where is a positive constant. Then Let Á ! 0 in (30). By (31)-(37), we have d dt by Gronwall Lemma, if we return to denote u as u K , we have By (38), from (25), it is easy to show that  (25) by 'S Á .u K k/ and integrating over Q T , as we have got (20), we obtain By Lemma 2.5, s; x; t /ds j 2 'dxdt: (41) Noticing that E D fE i g 2 .L 2 .Q T // N and divE 2 L 2 .Q T /, let K ! 1 in (40), we get (8).
The proof of (9) is similar to that in [19] et.al, we omit the details here.

The Proof of Theorem 2.3
Let u be the set of all jump points of u 2 BV .Q T /, v the normal of u at X D .x; t /, u C .X / and u .X/ the approximate limits of u at X 2 u with respect to .v; Y X / > 0 and .v; Y X / < 0 respectively. For continuous function p.u; x; t / and u 2 BV .
which is called the composite mean value of p and u. For a given t, we denote t u ; H t ; .v t 1 ; ; v t N / and u ṫ as all jump points of u. ; t /, Hausdorff measure of t u , the unit normal vector of t u , and the asymptotic limit of u. ; t/ respectively. By [17], if f .s/ 2 C 1 .R/; u 2 BV .Q T /, then f .u/ 2 BV .Q T / and where I.˛;ˇ/ denote the closed interval with endpoints˛andˇ. Proof.
x; t / > 0g: First prove a.s/ D 0; s 2 I.u C .x; t /; u .x; t // a.e. on 1 . Since any measurable subset of 1 can be expressed as the union of a Borel set and a set of measure zero, it suffices to prove a.s; x; t / D 0; s 2 I.u C .x; t /; u .x; t // a.e. on U 1 ; where U is a Borel subset of 1 . We may suppose U is compact. By Lemma 3.7.8 in [22], for any bounded function f .x; t /, which is measurable with respect to measure @u @x i , we have where U t D fx W .x; t / 2 U g. By [19], for any Borel subset S U , The definition of 1 implies that the left hand side vanishes, so we have x; t / u t .x; t // sgnv t i , where u .x; t / denote the characteristic function of U , sum up for i from 1 up to N . Then we obtain where G is the projection of U on the t-axis. Equality (46) implies for almost all t 2 G, and hence for almost all H t -almost everywhere on U t , which is impossible unless mesG D 0. For any˛;ˇwith 0 <˛<ˇ< T , we choose j .t / 2 C 1 0 .0; T / such that By [24], we can choose ' n 2 C 1 0 .Q T / such that j ' n .x; t / jÄ 1; lim Now from the definition of BV-function, we have Since mesG D 0, the three terms on the right hand vanish and Thus the lemma is proved.
Where .x; t / 0; .x; t / 2 C 1 0 .Q T /, and Since E i 2 L 2 .Q T / and 2 C 1 0 .Q T Q T /, by the control convergent theorem, we have Let h ! 0 in the above equality. We have .E ix i E iy i /vsgn.u v/ .x; t /j h .x y; t /dxdt dyd D 0: For the other terms in (52), i.e.