Global Existence of Classical Solutions for a Class Nonlinear Parabolic Equations

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1 Global Journal of Science Frontier Researc Matematics and Decision Sciences Volume 12 Issue 8 Version 1.0 Type : Double Blind Peer Reviewed International Researc Journal Publiser: Global Journals Inc. (USA Online ISSN: & Print ISSN: Global Eistence of Classical Solutions for a Class Nonlinear Parabolic Equations By Svetlin Georgiev Georgiev University of Sofia Abstract - In tis work we propose a new approac for investigating te local and global eistence of classical solutions of nonlinear parabolic equations. Tis approac gives new results. Keywords: nonlinear parabolic equation classical solutions local eistence global eistence. GJSFR-F Classification : AMS Subject Classification: 35K55 Global Eistence of Classical Solutions for a Class Nonlinear Parabolic Equations Strictly as per te compliance and regulations of :. Svetlin Georgiev Georgiev. Tis is a researc/review paper distributed under te terms of te Creative Commons Attribution-Noncommercial 3.0 Unported License ttp://creativecommons.org/licenses/by-nc/3.0/ permitting all non commercial use distribution and reproduction in any medium provided te original work is properly cited.

2 Ref. Global Eistence of Classical Solutions for a Class Nonlinear Parabolic Equations Svetlin Georgiev Georgiev 1. Fujita H. On te blowing up of solutions of te Caucy problem for ut = u + u1+ J. Fac. Sci. Univ. Tokyo Sect. I 13 ( Abstract - In tis work we propose a new approac for investigating te local and global eistence of classical solutions of nonlinear parabolic equations. Tis approac gives new results. Keywords and prases : nonlinear parabolic equation classical solutions local eistence global eistence. In tis paper we investigate te Caucy problem I. Introduction u t ( + mu = λ u p 1 u t [0 R n (1.1 u(0 = u 0 ( R n (1.2 were n 2 is fied = n λ m R p 1 u 0 ( C 2 (R n u 0 ( P (u 0 i ( P for every i = n and for every R n P > 0 is given constant. Here we propose new approac for invetsigating of tis problem wic gives new results. Te Caucy problem for te equation (1.1 is investigated by many autors. For instance see [1] and references terein. Obviously te problem for eistence of solutions to te Caucy problem (1.1 (1.2 is connected wit te Fujita s eponent wic depends of te dimension n and te values of te parameter p. Here we propose new conception about tis problem. Our tesis is tat tis problem depends only of te integral representation for te solutions wic is used. In tis work we propose new integral representation. Our conception tell us tat tere are cases in wic tere is a global eistence of solutions under specific set and local eistence and blow up under anoter specific set. We will illustrate our new conception wit te following eample. Eample. Let λ = 1 p = 3 m = 0. Ten is a solution to te problem Really for u(t = t+1 we ave u(t = t + 1 u t u = u 2 u t [0 R n u(0 = 1 2 R n. 1 u t = 2 2(t u t u = u 2 u 1 2 2(t = 1 1 2(t+1 2 t+1. Autor : University of Sofia Faculty of Matematics and Informatics Department of Differential Equations Blvd Tar Osvoboditel" 15 Sofia 1000 Bulgaria. sgg2000bg@yaoo.com (1.3 Journal of Science Frontier Researc F Volume XII Issue VIII V ersion I 1Global Global Journals Inc. (US

3 Global Eistence of Classical Solutions for a Class Nonlinear Parabolic Equations Tis eample and our main result we can consider as counter - eample of te well known teory. Our new nice result is due to our new integral representation. Tis approac is used for yperbolic equations in [2]. Our main result is Teorem 1.1. Let n 2 be fied p 1 be fied P > 0 be fied λ m R be fied u 0 ( C 2 (R n u 0 ( P (u 0 i ( P for every i = n and for every R n. Ten te Caucy problem (1.1 (1.2 as a solution u C 1 ([0 C 2 (R n. 2 Global Journal of Science Frontier Researc F Volume XII Issue VIII V ersion I Let ɛ is fied so tat 1 > ɛ > 0. For fied P > 0 we coose te constants A i i = n so tat λ (A 1 A 2... A n 2 P p 1 + (3 + m (A 1 A 2... A n 2 +(A 1 A 2... A n (A 1 A 2... A n 2 A n (A 2 A 3... A n 2 (1 ɛ ( λ (A 1 A 2 A j 1 A j+1 A n 2 P p 1 + (4 + m (A 1 A 2 A j 1 A j+1 A n 2 +(A 1 A 2 A j 1 A j+1 A n (A 2 A j 1 A j+1 A n 2 A j (1 ɛ j = n were A 0 = A n+1 = 1. Tere eist suc constants A 1 A 2... A n for wic te inequalities (2.1 are possible. For instance wen 1 > A i > 0 are enoug small i = n. Wit B 1 we will denote te set B 1 = R n : 0 i A i Firstly we will prove tat te Caucy problem i = n. (2.1 u t ( + mu = λ u p 1 u t [0 1] B 1 (2.2 u(0 = u 0 ( B 1 (2.3 as a solution u for wic u C 1 ([0 1] C 2 (B 1. For tis purpose we will use fied point arguments. Terefore we ave a need to define an operator wose fied points satisfy te above Caucy problem. Our observation is Lemma 2.1. If u C 1 ([0 1] C 2 (B 1 satisfies te integral equation λ t 0 u p 1 udsddy + m t 0 u(t sdsd + u 0(sdsd = 0 t [0 1] B 1 ten u is a solution to te Caucy problem (2.2 (2.3. Here u(y sdsddy t i=1 0 i i u(y ŝ i ds i d i dy s i = (s 1... s i 1 s i+1... s n ŝ i = (s 1... s i 1 i s i+1... s n = 1 1 n n i = 1 1 i 1 i 1 Proof. We differentiate in t te equality (2.4 and obtain λ u p 1 udsd + m u t(t sdsd = 0. II. Proof of Teorem 1.1 i+1 i+1 n n. u(t sdsd i=1 i i u(t ŝ i ds i d i (2.4 Ref. 2. Georgiev S. Global eistence for nonlinear wave equation. Dyn. PDE Vol. 7 No. 3 pp Global Journals Inc. (US

4 Global Eistence of Classical Solutions for a Class Nonlinear Parabolic Equations Now we differentiate twice in 1 after wic twice in 2 and etc. twice in n and we obtain We put t = 0 in (2.4 and we obtain u t ( + mu = λ u p 1 u. Ref. 3. Xiang T. Rong Yuan. A class of epansive - type Krasnosel'skii _ed point teorems. Nonlinear Analysis 71( u(0 sdsd + u 0 (sdsd = 0. After we differentiate te last equality twice in 1 after wic twice in 2 and etc. twice in n we obtain u 0 ( = u(0. Consequently u(t is a solution to te Caucy problem (2.2 (2.3. Te above lemma motivate us to define te integral operator L 11 (u = u(t + m t 0 t i=1 0 i i u(y ŝ i ds i d i dy u(y sdsddy + λ t 0 u(t s u 0 (s u p 1 udsddy dsd t [0 1] B 1. Our aim is to prove tat te operator L 11 as a fied point. We will use te following fied point teorem. Teorem 2.2. (see [3] Corrolary 2.4 pp Let X be a nonempty closed conve subset of a Banac space Y. Suppose tat T and S map X into Y suc tat (i S is continuous S(X resides in a compact subset of Y ; (ii T : X Y is epansive and onto. Ten tere eists a point X wit S + T =. Here we will use te following definition for epansive operator. Definition. (see [3] pp Let (X d be a metric space and M be a subset of X. Te mapping T : M X is said to be epansive if tere eists a constant > 1 suc tat d(t T y d( y y M. For tis purpose we will use te representation of te operator L 11 as follows were T 11 (u = (1 + ɛu(t S 11 (u = ɛu(t + m t 0 L 11 (u = T 11 (u + S 11 (u t i=1 0 i i u(y ŝ i ds i d i dy u(y sdsddy + λ t 0 u(t s u 0 (s u p 1 udsddy dsd. Also we define te sets M 11 = u(t C 1 ([0 1] C 2 u(t (B 1 ma t [01] ma B1 P ui ma t [01] ma B1 (t P N 11 = u(t C 1 ([0 1] C 2 (B 1 ui ma t [01] ma B1 (t (1 + ɛp i = n ma t [01] ma B1 u(t (1 + ɛp i = n Journal of Science Frontier Researc F Volume XII Issue VIII V ersion I 3Global Global Journals Inc. (US

5 4 Global Eistence of Classical Solutions for a Class Nonlinear Parabolic Equations were u 0 = u t. In tese sets we define a norm as follows u(t u 2 = sup : (t [0 1] B1. Lemma 2.3. Te sets M 11 and N 11 are closed compact and conve spaces in C([0 1] B 1 in te sense of norm 2. Proof. We will prove our assertion for M 11. Let u n is a sequence of elements of M 11 and u n n u in te sense of te norm 2. Evidently u C([0 1] B 1 and u(t P for every (t [0 1] B 1. We suppose tat u / C 1 ([0 1] B 1. Ten tere eists j n and ɛ > 0 so tat for every δ 1 = δ 1 (ɛ > 0 and < δ 1 0 ( j 1 j + j+1... n [0 1] B 1 we ave u( 0 1 j 1 j + j+1 n u( 0 1 n > ɛ. (2.5 Global Journal of Science Frontier Researc F Volume XII Issue VIII V ersion I On te oter and since u n C 1 ([0 1] C 2 (B 1 we ave tat tere eists δ 2 = δ 2 (ɛ > 0 so tat from < δ 2 0 ( j 1 j + j+1... n [0 1] B 1 we ave u n( 1 j 1 j + j+1 n u n ( 1 n < ɛ 3. (2.6 Also from u n n u in te sense of te norm 2 we ave for enoug large n and < minδ 1 δ 2 0 ( j 1 j + j+1 n [0 1] B 1 tat un(01 j 1j+j+1 n un(01 n u(01 j 1j+j+1 n u(01 n < ɛ 3. (2.7 Ten from (2.7 (2.6 (2.5 we obtain for < minδ 1 δ 2 0 ( j 1 j + j+1 n [0 1] B 1 for enoug large n ɛ < + u(01 j 1j+j+1 n u(01 n un(01 j 1j+j+1 n un(01 n un(01 j 1j+j+1 n un(01 n wic is a contradiction wit our assumption tat u / C 1 ([0 1] B 1. Terefore u C 1 ([0 1] B 1 u(01 j 1j+j+1 n u(01 n < 2 ɛ 3 We suppose tat u / C 1 ([0 1] C 2 (B 1. Ten tere eists j n and ɛ 1 > 0 so tat for every δ 3 = δ 3 (ɛ 1 > 0 and < δ 3 0 ( j 1 j + j+1... n [0 1] B 1 we ave u j ( 0 1 j 1 j + j+1 n u j ( 0 1 n > ɛ 1. (2.8 On te oter and since u n C 1 ([0 1] C 2 (B 1 we ave tat tere eists δ 4 = δ 4 (ɛ 1 > 0 so tat from < δ 2 0 ( j 1 j + j+1... n [0 1] B 1 we ave (u n j ( 0 1 j 1 j + j+1 n (u n j ( 0 1 n < ɛ 3. (2.9 Global Journals Inc. (US

6 Global Eistence of Classical Solutions for a Class Nonlinear Parabolic Equations Also from u n n u in te sense of te norm 2 we ave for enoug large n and < minδ 3 δ 4 0 ( 1 2 j 1 j + j+1 n B 1 tat (un j (01 j 1j+j+1 n (un (01 n j u j (01 j 1j+j+1 n u (01 n j < ɛ1 3. (2.10 Ten from (2.10 (2.9 (2.8 we obtain for < minδ 3 δ 4 0 ( 1 2 j 1 j + j+1 n B 1 for enoug large n ɛ 1 < + u j (01 j 1j+j+1 n u (01 n j (un j (01 j 1j+j+1 n (un (01 n j (un j (01 j 1j+j+1 n (un (01 n j u j (01 j 1j+j+1 n u (01 n j <2 ɛ1 3 wic is a contradiction wit our assumption tat u / C 1 ([0 1] C 2 (B 1. Terefore u C 1 ([0 1] C 2 (B 1. Now we suppose tat tere eists j n and ( t [0 1] B 1 so tat Ten tere eists ɛ 2 > 0 so tat u j ( t > P. u j ( t P + ɛ 2. From ere tere eists δ 5 = δ 5 (ɛ 2 > 0 suc tat from < δ 5 0 ( t 1... j 1 j + j+1... n [0 1] B 1 we ave u( t 1... j 1 j + j+1... n u( t P + ɛ 2. Journal of Science Frontier Researc F Volume XII Issue VIII V ersion I 5Global On te oter and since u n ( t u( t in sense of 2 as n follows tat tere eists δ 6 = δ 6 (ɛ 2 > 0 so tat we ave from < δ 6 0 ( t 1... j 1 j + j+1... n [0 1] B 1 un( t 1... j 1 j+ j+1... n u n( t u( t 1... j 1 j+ j+1... n u( t < ɛ 2 and since (u n j P in [0 1] B 1 u n( t 1... j 1 j + j+1... n u n ( t P for enoug large n. From ere for enoug large n and for < minδ 5 δ 6 0 ( t 1... j 1 j + j+1... n [0 1] B 1 we ave ɛ 2 = P + ɛ 2 P u( t 1... j 1 j+ j+1... n u( t u( t 1... j 1 j+ j+1... n u( t un( t 1... j 1 j+ j+1... n u n( t un( t 1... j 1 j+ j+1... n u n( t < ɛ 2 wic is a contradiction. Terefore u j P in [0 1] B 1 for every j = n. Consequently u M 11 and M 11 is closed in C([0 1] B 1 in sense of 2. Using Arela - Ascoli Teorem te set M 11 is a compact set in C([0 1] B 1 in sense of 2. Global Journals Inc. (US

7 Global Eistence of Classical Solutions for a Class Nonlinear Parabolic Equations 6 Global Journal of Science Frontier Researc F Volume XII Issue VIII V ersion I Let now λ [0 1] is arbitrary cosen and fied and u 1 u 2 M 11. Ten for (t [0 1] B 1 we ave λu 1 (t + (1 λu 2 (t C 1 ([0 1] C 2 (B 1 and u i (t P u ij (t P for j = n i = 1 2 λu 1 (t + (1 λu 2 (t λ u 1 (t + (1 λ u 2 (t λp + (1 λp = P λu 1j (t + (1 λu 2j (t λ u 1j (t + (1 λ u 2j (t λp + (1 λp = P j = n. Terefore M 11 is conve. As in above we can prove tat N 11 is closed compact and conve in C([0 1] B 1 in sense of 2. Lemma 2.4. Te operator T 11 : M 11 N 11 is an epansive operator and onto. Proof. Let u M 11. Ten u C 1 ([0 1] C 2 (B 1 from ere (1 + ɛu C 1 ([0 1] C 2 (B 1 i.e. T 11 (u C 1 ([0 1] C 2 (B 1 and T 11 (u (1 + ɛp (T 11 (u i (1 + ɛp i = n. Let now u v M 11. Ten we ave T 11 (u T 11 (v 2 = (1 + ɛ u v 2. From ere and above follows tat T 11 : M 11 N 11 is an epansive operator. Now we will see tat T 11 : M 11 N 11 is onto. Really let v N 11 v 0. Let also u = From ere ma t [01] ma B1 u P ma t [01] ma B1 u i P for i = n. Terefore u M 11 and te operator T 11 : M 11 N 11 is onto. Lemma 2.5. We ave is continuous. S 11 : M 11 M 11 Proof. Let u M 11 is arbitrary cosen element. Ten using te definition of te operator S 11 we ave Also S 11 (u ɛ u(t + m t 0 t i=1 0 i i u(y ŝ i ds i d i dy + u(y s dsddy + λ t 0 u(t s + u 0 (s u p dsddy dsd ɛp + m (A 1 A 2... A n 2 P + λ (A 1 A 2... A n 2 P p + 3(A 1 A 2... A n 2 P +P ((A 1 A 2... A n (A 1 A 2... A n 2 A n (A 2 A 3... A n 2 ɛp + (1 ɛp = P. (S 11 (u t = ɛu t (t + m Ten u t(t sdsd. u(t sdsd + λ (S 11 (u t ɛ u t (t + m i=1 i i u(t ŝ i ds i d i + v 1+ɛ. u p 1 udsd i=1 i i u(t ŝ i ds i d i u(t s dsd + λ u t(t s dsd ɛp + λ (A 1 A 2... A n 2 P p + (2 + m (A 1 A 2... A n 2 P u p dsd +P ((A 1 A 2... A n (A 1 A 2... A n 2 A n (A 2 A 3... A n 2 ɛp + (1 ɛp = P. Global Journals Inc. (US

8 Global Eistence of Classical Solutions for a Class Nonlinear Parabolic Equations Let now j = 1 2 n is arbitrary cosen and fied. Ten ( S 11 (u = ɛu j (t m t 0 j ẑ j u(y sdsd j dy λ t 0 j ẑ j u p 1 udsd j dy j were and from ere ( S 11 (u n t i=1i j 0 ij ẑ ij + A j ẑ j (u(t ŝ j u 0 (ŝ j dŝ j d j u(y ŝ i ds i d i dy + t 0 j j u j (y ŝ j ds j d j dy ij = 1 1 i 1 i+1 i 1 i+1 j 1 j+1 j 1 j+1 n n ẑ j = 1 1 j ɛ uj (t + m t 0 t i=1i j 0 ij ẑ ij + A j ẑ j ( u(t ŝ j + u 0 (ŝ j dŝ j d j j 1 j 1 j j+1 j j+1 n n j ẑ j u(y s dsd j dy + λ t 0 j ẑ j u p dsd j dy u(y ŝ i ds i d i dy + t 0 j j u j (y ŝ j ds j d j dy Journal of Science Frontier Researc Volume XII Issue V ersion I 7Global ɛp + λ (A 1 A 2 A j 1 A j+1 A n 2 A j P p + (4 + m (A 1 A 2 A j 1 A j+1 A n 2 A j P + ((A 1 A 2 A j 1 A j+1 A n (A 2 A j 1 A j+1 A n 2 A j P ɛp + (1 ɛp = P. VIII Consequently S 11 : M 11 M 11. From te above estimates for S 11 (u (S 11 (u j j = n follows tat if v n n v in te sense of te topology of te set M 11 v n v M 11 we ave tat S 11 (v n n S 11 (v in te sense of topology of te set M 11. Terefore te operator S 11 : M 11 M 11 is a continuous operator. Using Lemma 2.3 Lemma 2.4 Lemma 2.5 we apply Teorem 2.2 as te operator S in Teorem 2.2 corresponds of S 11 te operator T in Teorem 2.2 corresponds of T 11 te set X in Teorem 2.2 corresponds of M 11 and Y in Teorem 2.2 corresponds of N 11 and terefore follows tat tere eists u 11 M 11 so tat u 11 = S(u 11 + T (u 11 i.e. u 11 is a fied point of te operator L 11. From ere and Lemma 2.1 follows tat u 11 is a solution to te Caucy problem (2.2 (2.3 for wic u 11 C 1 ([0 1] C 2 (B 1. F Now we define te set B 2 = R n : A 1 1 2A 1 0 i A i i = 2... n te operators L 12 (u = u(t + m t 0 t i=2 0 i i u(y ŝ i ds i d i dy ( u(t s u 11 (0 s dsd u(y sdsddy + λ t 0 u p 1 udsddy + t (u(y 1 s 2... s n u 11 (y A 1 s 2... s n + (A 1 1 u 11 1 (y A 1 s 2... s n ds 1 d 1 dy t [0 1] B 2 L 12 (u = T 12 (u + S 12 (u Global Journals Inc. (US

9 were Global Eistence of Classical Solutions for a Class Nonlinear Parabolic Equations T 12 (u = (1 + ɛu(t t [0 1] B 2 S 12 (u = ɛu(t + m t 0 u(y sdsddy + λ t 0 u p 1 udsddy t i=2 0 i i u(y ŝ i ds i d i dy u(t s u 11 (0 s dsd + t (u(y 1 s 2... s n u 11 (y A 1 s 2... s n + (A 1 1 u 11 1 (y A 1 s 2... s n ds 1 d 1 dy t [0 1] B 2 8 Global Journal of Science Frontier Researc F Volume XII Issue VIII V ersion I te sets M 12 = u(t C 1 ([0 1] C 2 (B 2 ma t [01] ma B2 u(t P ui ma t [01] ma B2 (t P i = n N 12 = u(t C 1 ([0 1] C 2 u(t (B 2 ma t [01] ma B2 (1 + ɛp ui ma t [01] ma B2 (t (1 + ɛp i = n in tese sets we define a norm as follows u(t u 2 = sup : (t [0 1] B2. Te sets M 12 and N 12 are closed conve and compact in C([0 1] B 2 in sense of 2. As in above we conclude tat tere eists u 12 M 12 so tat L 12 u 12 = u 12 i.e. u 12 is a solution to te Caucy problem For u 12 we ave m t 0 u12 (y sdsddy + λ t 0 t i=2 0 i i u 12 (y ŝ i ds i d i dy u 12 (t s u 11 (0 s u t ( + mu = λ u p 1 u t [0 1] B 2 u(0 = u 11 (0 B 2. dsd u12 p 1 u 12 dsddy + t (u 12 (y 1 s 2... s n u 11 (y A 1 s 2... s n + (A 1 1 u 11 1 (y A 1 s 2... s n ds 1 d 1 dy = 0. (2.11 Now we put 1 = A 1 in te last equality and we obtain t (u 12 (y A 1 s 2... s n u 11 (y A 1 s 2... s n ds 1 d 1 dy = 0 and we differentiate it in t twice in 2 and etc. twice in n and we obtain Global Journals Inc. (US u 12 1 =A 1 = u 11 1 =A 1. Now we differentiate in 1 te equality (2.11 after wic we put 1 = A 1 and we obtain t (u 12 1 (y A 1 s 2... s n u 11 1 (y A 1 s 2... s n ds 1 d 1 dy = 0

10 9Global Journal of Science Frontier Researc Volume XII Issue ersion I Global Eistence of Classical Solutions for a Class Nonlinear Parabolic Equations and we differentiate it in t twice in 2 and etc. twice in n and we obtain Using te equalities (u =A 1 = (u =A 1. u 11 t ( + mu 11 = λ u 11 p 1 u 11 u 12 t ( + mu 12 = λ u 12 p 1 u 12 u 11 1 =A 1 = u 12 1 =A 1 (u =A 1 = (u =A 1 we conclude tat (u =A 1 = (u =A 1. In tis way we obtain tat te function u 11 t [0 1] 0 1 A A n A n u = u 12 t [0 1] A 1 1 2A A n A n is a solution to te Caucy problem u t ( + mu = λ u p 1 u t [0 1] 0 1 2A A n A n u(0 = u 0 ( 0 1 2A A n A n. Repeat te above steps in 1 2 and etc. n we obtain a solution u 1 to te Caucy problem (1.1 (1.2 wic belongs to te space C 1 ([0 1] C 2 (R n. Now we consider te Caucy problem u t ( + mu = λ u p 1 u t [1 2] B 1 u(1 = u 1 (1 B 1. (2.12 V VIII F For tis purpose we consider te operator L 1 11(u = T 1 11(u + S 1 11(u were T 1 11(u = (1 + ɛu(t S11(u 1 = ɛu(t + m t 1 t i=1 1 i i u(y ŝ i ds i d i dy u(y sdsddy + λ t 1 u p 1 udsddy ( A u(t s u 1 (1 s dsd. Also we define te sets M11 1 = u(t C 1 ([1 2] C 2 u(t (B 1 ma t [12] ma B1 P ui ma t [12] ma B1 (t P N 1 11 = u(t C 1 ([1 2] C 2 (B 1 ui ma t [12] ma B1 (t (1 + ɛp i = n ma t [12] ma B1 u(t (1 + ɛp i = n. In tese sets we define a norm as follows u(t u 2 = sup : (t [1 2] B1 Global Journals Inc. (US

11 Global Eistence of Classical Solutions for a Class Nonlinear Parabolic Equations tese sets are closed conve and compact in C([1 2] B 1 in sense of. As in above we conclude tat te problem (2.12 as a solution u 11 1 C 1 ([1 2] C 2 (B 1. Now we consider te Caucy problem u t ( + mu = λ u p 1 u t [1 2] B 2 u(1 = u 1 (1 (2.13 L 1 12(u = u(t + m t 1 t i=2 1 i i u(y ŝ i ds i d i dy u(t s u 1 (1 s dsd u(y sdsddy + λ t 1 u p 1 udsddy 10 Global Journal of Science Frontier Researc F Volume XII Issue VIII V ersion I + t (u(y 1 s 2... s n u 11 1 (y A 1 s 2... s n + (A 1 1 u 11 1 (y A 1 s 2... s n ds 1 d 1 dy t [1 2] B 2 were T 1 12(u = (1 + ɛu(t t [1 2] B 2 S12(u 1 = ɛu(t + m t 1 t i=2 1 i i u(y ŝ i ds i d i dy u(t s u 1 (1 s dsd L 1 12(u = T 1 12(u + S 1 12(u u(y sdsddy + λ t 1 u p 1 udsddy + t (u(y 1 s 2... s n u 11 1 (y A 1 s 2... s n + (A 1 1 u (y A 1 s 2... s n ds 1 d 1 dy t [1 2] B 2 te sets M 1 12 = u(t C 1 ([1 2] C 2 (B 2 ma t [12] ma B2 u(t P ui ma t [12] ma B2 (t P N 1 12 = u(t C 1 ([1 2] C 2 (B 2 ui ma t [12] ma B2 (t (1 + ɛp i = n ma t [12] ma B2 u(t (1 + ɛp i = n in tese sets we define a norm as follows u(t u 2 = sup : (t [1 2] B2. Te sets M 1 12 and N 1 12 are closed conve and compact in C([1 2] B 2 in sense of 2. As in te step 1 we conclude tat te problem (2.13 as a solution u 12 1 C 1 ([1 2] C 2 (B 2. Since u 12 1 (1 = u 1 (1 for B 2 and u 11 1 (1 = u 1 (1 for B 1 we ave tat u 12 1 t=11 =A 1 = u 1 t=11 =A 1 (u t=11 =A 1 = (u 1 1 t=11 =A 1 Global Journals Inc. (US

12 Global Eistence of Classical Solutions for a Class Nonlinear Parabolic Equations (u t=11 =A 1 = (u t=11 =A 1 u 11 1 t=11 =A 1 = u 1 t=11 =A 1 (2.14 (u t=11 =A 1 = (u 1 1 t=11 =A 1 (u t=11 =A 1 = (u t=11 =A 1 as in te case t [0 1] we ave u =A 1 = u =A 1 (u =A 1 = (u =A 1 (u =A 1 = (u =A 1 (2.15 for t [1 2] and etc. In tis way we obtain a solution u 2 C 1 ([1 2] C 2 (R n to te Caucy problem 11 u t ( + mu = λ u p 1 u t [1 2] R n u(1 = u 1 (1 R n. Using reasonings as (2.14 (2.15 we ave tat u 1 t [0 1] R n is a solution to te Caucy problem u 2 t [1 2] R n u t ( + mu = λ u p 1 u t [0 2] R n u(0 = u 0 ( R n wic belongs to te space C 1 ([0 2] C 2 (R n and etc. we obtain a solution to te problem (1.1 (1.2 wic belongs to te space C 1 ([0 C 2 (R n. III. Acknowledgements Tis article is supported by te grant DD-VU 02/90 Bulgaria. References Références Referencias 1. Fujita H. On te blowing up of solutions of te Caucy problem for ut = u + u1+ J. Fac. Sci. Univ. Tokyo Sect. I 13 ( Georgiev S. Global eistence for nonlinear wave equation. Dyn. PDE Vol. 7 No. 3 pp Xiang T. Rong Yuan. A class of epansive - type Krasnosel'skii _ed point teorems. Nonlinear Analysis 71( Global Journal of Science Frontier Researc F Volume XII Issue VIII V ersion I Global Journals Inc. (US

13 Global Eistence of Classical Solutions for a Class Nonlinear Parabolic Equations 12 Global Journal of Science Frontier Researc F Volume XII Issue VIII V ersion I Tis page is intentionally left blank Global Journals Inc. (US